Integrand size = 24, antiderivative size = 882 \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=-\frac {c \sqrt {c+a^2 c x^2}}{30 a^3}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{60 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^2}+\frac {1}{20} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {31 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{240 a^3}-\frac {19 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{120 a}-\frac {1}{10} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {c x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{16 a^2}+\frac {7}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {41 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {41 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {41 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}} \]
-1/60*(a^2*c*x^2+c)^(3/2)/a^3+1/8*I*c^2*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)+3/8*I*c^2*polylo g(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)+3/16*I*c^2*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)+41/120*I*c^2*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)-3/8*I*c^2* 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)-3/16*I*c^2*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)+3/8*c^2*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)- 3/8*c^2*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)+41/60*I*c^2*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)-41/120*I*c^2* 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)-1/30*c*(a^2*c*x^2+c)^(1/2)/a^3+1/12*c*x*arctan(a*x)*(a^2*c*x ^2+c)^(1/2)/a^2+1/20*c*x^3*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+31/240*c*arctan (a*x)^2*(a^2*c*x^2+c)^(1/2)/a^3-19/120*c*x^2*arctan(a*x)^2*(a^2*c*x^2+c)^( 1/2)/a-1/10*a*c*x^4*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+1/16*c*x*arctan(a*x) ^3*(a^2*c*x^2+c)^(1/2)/a^2+7/24*c*x^3*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+1/ 6*a^2*c*x^5*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4015\) vs. \(2(882)=1764\).
Time = 18.47 (sec) , antiderivative size = 4015, normalized size of antiderivative = 4.55 \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\text {Result too large to show} \]
(c*((Sqrt[c*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + ( Sqrt[c*(1 + a^2*x^2)]*(-(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLo g[2, I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2) ]*(-1/8*(Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]]) - (3*Pi^2*((Pi/2 - ArcTan[ a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a *x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi /2 - ArcTan[a*x]))])))/4 + (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi /2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*( Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + Pol yLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 - ArcTan[a*x])^ 4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*L og[1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan [a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (P i/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + Arc Tan[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (P i/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTa n[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x...
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \int x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3dx+a^2 c \int x^4 \sqrt {a^2 c x^2+c} \arctan (a x)^3dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \left (c \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+a^2 c \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (a^2 c \int \frac {x^6 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )+a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )\right )+a^2 c \left (c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+a^2 c \left (-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}+\frac {x^5 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{6 a^2 c}\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )+a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )\right )+a^2 c \left (c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+a^2 c \left (-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}+\frac {x^5 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{6 a^2 c}\right )\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle a^2 c \left (c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+a^2 c \left (-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}+\frac {x^5 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{6 a^2 c}\right )\right )+c \left (a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c \left (c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+a^2 c \left (-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}+\frac {x^5 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{6 a^2 c}\right )\right )+c \left (a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle a^2 c \left (c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+a^2 c \left (-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}+\frac {x^5 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{6 a^2 c}\right )\right )+c \left (a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle a^2 c \left (c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+a^2 c \left (-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}+\frac {x^5 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{6 a^2 c}\right )\right )+c \left (a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle a^2 c \left (c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+a^2 c \left (-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}+\frac {x^5 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{6 a^2 c}\right )\right )+c \left (a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+a^2 c \left (-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}+\frac {x^5 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{6 a^2 c}\right )\right )+c \left (a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle a^2 c \left (c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+a^2 c \left (-\frac {\int \frac {x^5 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {5 \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}+\frac {x^5 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{6 a^2 c}\right )\right )+c \left (a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 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 )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle 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} \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 )}{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 )\) |
3.5.21.3.1 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 = 4.10 (sec) , antiderivative size = 514, normalized size of antiderivative = 0.58
| method | result | size |
| default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (40 \arctan \left (a x \right )^{3} a^{5} x^{5}-24 a^{4} \arctan \left (a x \right )^{2} x^{4}+70 \arctan \left (a x \right )^{3} a^{3} x^{3}+12 \arctan \left (a x \right ) x^{3} a^{3}-38 x^{2} \arctan \left (a x \right )^{2} a^{2}+15 \arctan \left (a x \right )^{3} a x -4 a^{2} x^{2}+20 x \arctan \left (a x \right ) a +31 \arctan \left (a x \right )^{2}-12\right )}{240 a^{3}}+\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-45 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+45 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+82 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-82 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-82 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+82 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{240 a^{3} \sqrt {a^{2} x^{2}+1}}\) | \(514\) |
1/240*c/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(40*arctan(a*x)^3*a^5*x^5-24*a^4*arc tan(a*x)^2*x^4+70*arctan(a*x)^3*a^3*x^3+12*arctan(a*x)*x^3*a^3-38*x^2*arct an(a*x)^2*a^2+15*arctan(a*x)^3*a*x-4*a^2*x^2+20*x*arctan(a*x)*a+31*arctan( a*x)^2-12)+1/240*c*(c*(a*x-I)*(I+a*x))^(1/2)*(15*arctan(a*x)^3*ln(1+I*(1+I *a*x)/(a^2*x^2+1)^(1/2))-15*arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/ 2))-45*I*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+45*I*arct an(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+82*arctan(a*x)*ln(1+I*( 1+I*a*x)/(a^2*x^2+1)^(1/2))+90*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2 +1)^(1/2))-82*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-90*arctan(a* x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+90*I*polylog(4,-I*(1+I*a*x)/(a ^2*x^2+1)^(1/2))-90*I*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-82*I*dilog( 1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+82*I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2 )))/a^3/(a^2*x^2+1)^(1/2)
\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]
\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]