Integrand size = 24, antiderivative size = 1019 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\frac {13 c^2 \sqrt {c+a^2 c x^2}}{6720 a^3}-\frac {3 c \left (c+a^2 c x^2\right )^{3/2}}{560 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{280 a^3}+\frac {43 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{1344 a^2}+\frac {29}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1373 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{13440 a^3}-\frac {737 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{6720 a}-\frac {83}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{128 a^2}+\frac {59}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {17}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {397 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 )}{840 a^3 \sqrt {c+a^2 c x^2}}-\frac {15 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}-\frac {397 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 )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {397 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 )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {15 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {15 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {15 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {15 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}} \]
-3/560*c*(a^2*c*x^2+c)^(3/2)/a^3-1/280*(a^2*c*x^2+c)^(5/2)/a^3+15/64*I*c^3 *polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^ 2+c)^(1/2)+5/64*I*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^3*(a ^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-15/64*I*c^3*polylog(4,I*(1+I*a*x)/ (a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+397/840*I*c^3 *arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^3 /(a^2*c*x^2+c)^(1/2)+15/128*I*c^3*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2 *x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-397/1680*I*c^3*po lylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x ^2+c)^(1/2)+15/64*c^3*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2) )*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-15/64*c^3*arctan(a*x)*polylog( 3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2) +397/1680*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^3/(a^2*c*x^2+c)^(1/2)-15/128*I*c^3*arctan(a*x)^2*polylog(2,-I*(1+I* a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+13/6720* c^2*(a^2*c*x^2+c)^(1/2)/a^3+43/1344*c^2*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/ a^2+29/560*c^2*x^3*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+1/56*a^2*c^2*x^5*arctan (a*x)*(a^2*c*x^2+c)^(1/2)+1373/13440*c^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2) /a^3-737/6720*c^2*x^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a-83/560*a*c^2*x^4 *arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)-3/56*a^3*c^2*x^6*arctan(a*x)^2*(a^2*...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(6517\) vs. \(2(1019)=2038\).
Time = 24.69 (sec) , antiderivative size = 6517, normalized size of antiderivative = 6.40 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\text {Result too large to show} \]
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.
| \(\displaystyle \int x^2 \arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \int x^2 \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3dx+a^2 c \int x^4 \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \left (c \int x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3dx+a^2 c \int x^4 \sqrt {a^2 c x^2+c} \arctan (a x)^3dx\right )+a^2 c \left (a^2 c \int x^6 \sqrt {a^2 c x^2+c} \arctan (a x)^3dx+c \int x^4 \sqrt {a^2 c x^2+c} \arctan (a x)^3dx\right )\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \left (c \left (c \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+a^2 c \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (a^2 c \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx\right )\right )+a^2 c \left (a^2 c \left (a^2 c \int \frac {x^8 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \int \frac {x^7 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a}-\frac {7 \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \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 )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\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}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \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 )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \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 )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
3.5.29.3.1 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[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
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[1/(c*Sqrt[d]) Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[ c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Gt Q[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 = 12.63 (sec) , antiderivative size = 566, normalized size of antiderivative = 0.56
| method | result | size |
| default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (1680 \arctan \left (a x \right )^{3} a^{7} x^{7}-720 a^{6} x^{6} \arctan \left (a x \right )^{2}+4760 \arctan \left (a x \right )^{3} a^{5} x^{5}+240 \arctan \left (a x \right ) a^{5} x^{5}-1992 a^{4} \arctan \left (a x \right )^{2} x^{4}+4130 \arctan \left (a x \right )^{3} a^{3} x^{3}-48 a^{4} x^{4}+696 \arctan \left (a x \right ) x^{3} a^{3}-1474 x^{2} \arctan \left (a x \right )^{2} a^{2}+525 \arctan \left (a x \right )^{3} a x -168 a^{2} x^{2}+430 x \arctan \left (a x \right ) a +1373 \arctan \left (a x \right )^{2}-94\right )}{13440 a^{3}}+\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (525 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-525 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-1575 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+1575 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3176 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3176 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3176 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3176 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{13440 a^{3} \sqrt {a^{2} x^{2}+1}}\) | \(566\) |
1/13440*c^2/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(1680*arctan(a*x)^3*a^7*x^7-720* a^6*x^6*arctan(a*x)^2+4760*arctan(a*x)^3*a^5*x^5+240*arctan(a*x)*a^5*x^5-1 992*a^4*arctan(a*x)^2*x^4+4130*arctan(a*x)^3*a^3*x^3-48*a^4*x^4+696*arctan (a*x)*x^3*a^3-1474*x^2*arctan(a*x)^2*a^2+525*arctan(a*x)^3*a*x-168*a^2*x^2 +430*x*arctan(a*x)*a+1373*arctan(a*x)^2-94)+1/13440*c^2*(c*(a*x-I)*(I+a*x) )^(1/2)*(525*arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-525*arctan( a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-1575*I*arctan(a*x)^2*polylog(2, -I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+1575*I*arctan(a*x)^2*polylog(2,I*(1+I*a*x) /(a^2*x^2+1)^(1/2))+3176*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3 150*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3176*arctan(a*x) *ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150*arctan(a*x)*polylog(3,I*(1+I*a*x )/(a^2*x^2+1)^(1/2))+3150*I*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150 *I*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3176*I*dilog(1+I*(1+I*a*x)/(a^ 2*x^2+1)^(1/2))+3176*I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^3/(a^2*x^ 2+1)^(1/2)
\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]
\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]