Integrand size = 24, antiderivative size = 578 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=-\frac {115 c^2 \sqrt {c+a^2 c x^2}}{4032 a^4}-\frac {115 c \left (c+a^2 c x^2\right )^{3/2}}{18144 a^4}-\frac {23 \left (c+a^2 c x^2\right )^{5/2}}{7560 a^4}+\frac {\left (c+a^2 c x^2\right )^{7/2}}{252 a^4 c}+\frac {47 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{1344 a^3}-\frac {205 c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{6048 a}-\frac {103 a c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)}{1512}-\frac {1}{36} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{63 a^4}+\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{63 a^2}+\frac {5}{21} c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {19}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {115 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2016 a^4 \sqrt {c+a^2 c x^2}}+\frac {115 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {c+a^2 c x^2}}-\frac {115 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4032 a^4 \sqrt {c+a^2 c x^2}} \]
-115/18144*c*(a^2*c*x^2+c)^(3/2)/a^4-23/7560*(a^2*c*x^2+c)^(5/2)/a^4+1/252 *(a^2*c*x^2+c)^(7/2)/a^4/c+115/4032*I*c^3*polylog(2,-I*(1+I*a*x)^(1/2)/(1- I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/2016*I*c^3*arc tan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^ 2*c*x^2+c)^(1/2)-115/4032*I*c^3*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2 ))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-115/4032*c^2*(a^2*c*x^2+c)^(1 /2)/a^4+47/1344*c^2*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^3-205/6048*c^2*x^3 *arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a-103/1512*a*c^2*x^5*arctan(a*x)*(a^2*c*x ^2+c)^(1/2)-1/36*a^3*c^2*x^7*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-2/63*c^2*arct an(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^4+1/63*c^2*x^2*arctan(a*x)^2*(a^2*c*x^2+c) ^(1/2)/a^2+5/21*c^2*x^4*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+19/63*a^2*c^2*x^ 6*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+1/9*a^4*c^2*x^8*arctan(a*x)^2*(a^2*c*x ^2+c)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1320\) vs. \(2(578)=1156\).
Time = 6.72 (sec) , antiderivative size = 1320, normalized size of antiderivative = 2.28 \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx =\text {Too large to display} \]
((c + a^2*c*x^2)^(5/2)*(-48384*(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a* x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110* ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[a* x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTan[a*x]*Cos[5* ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^( I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log [1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^ (I*ArcTan[a*x])] - ((176*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x ^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*ArcTan[a*x]]) + 576*(1 + a^2*x^2)*(4116 + 10944*ArcTan[a*x]^2 + 6262*Cos[2*ArcTan[a*x]] - 5376*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 2764*Cos[4*ArcTan[a*x]] + 6720*Arc Tan[a*x]^2*Cos[4*ArcTan[a*x]] + 618*Cos[6*ArcTan[a*x]] - (10815*ArcTan[a*x ]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 6489*ArcTan[a*x]*Cos[3 *ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 2163*ArcTan[a*x]*Cos[5*ArcTan [a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*L og[1 - I*E^(I*ArcTan[a*x])] + (10815*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x ])])/Sqrt[1 + a^2*x^2] + 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^( I*ArcTan[a*x])] + 2163*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTa n[a*x])] + 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int x^5 \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2dx+c \int x^3 \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \left (a^2 c \int x^7 \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+c \int x^5 \sqrt {a^2 c x^2+c} \arctan (a x)^2dx\right )+c \left (a^2 c \int x^5 \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+c \int x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2dx\right )\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \left (a^2 c \left (a^2 c \int \frac {x^9 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx\right )\right )+c \left (a^2 c \left (a^2 c \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \int \frac {x^8 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{9 a}-\frac {8 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \int \frac {x^8 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{9 a}-\frac {8 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \int \frac {x^8 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{9 a}-\frac {8 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \int \frac {x^8 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{9 a}-\frac {8 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\int \frac {x^7}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{3 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\int \frac {x^7}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\int \frac {x^6}{\sqrt {a^2 c x^2+c}}dx^2}{16 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\int \left (\frac {\left (a^2 c x^2+c\right )^{5/2}}{a^6 c^3}-\frac {3 \left (a^2 c x^2+c\right )^{3/2}}{a^6 c^2}+\frac {3 \sqrt {a^2 c x^2+c}}{a^6 c}-\frac {1}{a^6 \sqrt {a^2 c x^2+c}}\right )dx^2}{16 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}+\frac {1}{a^4 \sqrt {a^2 c x^2+c}}\right )dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}+\frac {1}{a^4 \sqrt {a^2 c x^2+c}}\right )dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}+\frac {1}{a^4 \sqrt {a^2 c x^2+c}}\right )dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \int \frac {x^6 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{3 a}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{3 a}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx}{6 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}+\frac {1}{a^4 \sqrt {a^2 c x^2+c}}\right )dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}+\frac {1}{a^4 \sqrt {a^2 c x^2+c}}\right )dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right 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x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right 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)}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c 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a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^8}{9 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^7}{8 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^8 c^4}-\frac {6 \left (a^2 c x^2+c\right )^{5/2}}{5 a^8 c^3}+\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{a^8 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^8 c}}{16 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right )}{9 a}-\frac {8 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}\right )}{7 a^2}\right )}{9 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c 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a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right )}{7 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\right )\right )\) |
3.4.23.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/ (c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c *x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I *c*x])]/(c*Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan[c*x])^ p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] & & IGtQ[p, 0] && !GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b* ArcTan[c*x])^p/(c^2*d*m)), x] + (-Simp[b*f*(p/(c*m)) Int[(f*x)^(m - 1)*(( a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[f^2*((m - 1)/(c^ 2*m)) Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]
Time = 10.11 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.53
| method | result | size |
| default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (20160 \arctan \left (a x \right )^{2} a^{8} x^{8}-5040 \arctan \left (a x \right ) a^{7} x^{7}+54720 a^{6} x^{6} \arctan \left (a x \right )^{2}+720 a^{6} x^{6}-12360 \arctan \left (a x \right ) a^{5} x^{5}+43200 a^{4} \arctan \left (a x \right )^{2} x^{4}+1608 a^{4} x^{4}-6150 \arctan \left (a x \right ) x^{3} a^{3}+2880 x^{2} \arctan \left (a x \right )^{2} a^{2}-94 a^{2} x^{2}+6345 x \arctan \left (a x \right ) a -5760 \arctan \left (a x \right )^{2}-6157\right )}{181440 a^{4}}-\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{4032 a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(309\) |
1/181440*c^2/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*(20160*arctan(a*x)^2*a^8*x^8-50 40*arctan(a*x)*a^7*x^7+54720*a^6*x^6*arctan(a*x)^2+720*a^6*x^6-12360*arcta n(a*x)*a^5*x^5+43200*a^4*arctan(a*x)^2*x^4+1608*a^4*x^4-6150*arctan(a*x)*x ^3*a^3+2880*x^2*arctan(a*x)^2*a^2-94*a^2*x^2+6345*x*arctan(a*x)*a-5760*arc tan(a*x)^2-6157)-115/4032*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(arctan(a*x)*ln(1+ I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1 /2))-I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*dilog(1-I*(1+I*a*x)/(a^2*x ^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)
\[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2} \,d x } \]
\[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
\[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2} \,d x } \]
Exception generated. \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]