Integrand size = 24, antiderivative size = 675 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=-\frac {a^2 c^2 \sqrt {c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}-\frac {2 a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}-\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {26 a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+a^3 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {13 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {13 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {5 a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {5 a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
-1/3*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2/x^3+a^3*c^(5/2)*arctanh(a*x*c^(1/ 2)/(a^2*c*x^2+c)^(1/2))-5*I*a^3*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*ar ctan(a*x)^2*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-26/3*a^3*c^3*arctan(a*x) *arctanh((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^ (1/2)+5*I*a^3*c^3*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a ^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-5*I*a^3*c^3*arctan(a*x)*polylog(2,I*(1 +I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+13/3*I*a^ 3*c^3*polylog(2,-(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c *x^2+c)^(1/2)-13/3*I*a^3*c^3*polylog(2,(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a ^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-5*a^3*c^3*polylog(3,-I*(1+I*a*x)/(a^2* x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+5*a^3*c^3*polylog(3,I* (1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-1/3*a^2 *c^2*(a^2*c*x^2+c)^(1/2)/x-a^3*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-1/3*a*c ^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/x^2-2*a^2*c^2*arctan(a*x)^2*(a^2*c*x^2+ c)^(1/2)/x+1/2*a^4*c^2*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)
Time = 2.94 (sec) , antiderivative size = 644, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=-\frac {c^3 \sqrt {1+a^2 x^2} \left (2 \left (1+a^2 x^2\right )^{3/2}+12 a^3 x^3 \sqrt {1+a^2 x^2} \arctan (a x)+24 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)^2-6 a^4 x^4 \sqrt {1+a^2 x^2} \arctan (a x)^2+4 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2+12 i a^3 x^3 \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-12 a^3 x^3 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-2 \left (1+a^2 x^2\right )^{3/2} \cos (2 \arctan (a x))-3 a x \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-51 a^3 x^3 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-24 a^3 x^3 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+24 a^3 x^3 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )+3 a x \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+51 a^3 x^3 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )-52 i a^3 x^3 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-60 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+60 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+52 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+60 a^3 x^3 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-60 a^3 x^3 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )+2 \left (1+a^2 x^2\right )^{3/2} \arctan (a x) \sin (2 \arctan (a x))+\left (1+a^2 x^2\right )^{3/2} \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right ) \sin (3 \arctan (a x))-\left (1+a^2 x^2\right )^{3/2} \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right ) \sin (3 \arctan (a x))\right )}{12 x^3 \sqrt {c+a^2 c x^2}} \]
-1/12*(c^3*Sqrt[1 + a^2*x^2]*(2*(1 + a^2*x^2)^(3/2) + 12*a^3*x^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 24*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - 6*a^4* x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 4*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2 + (12*I)*a^3*x^3*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 12*a^3*x^3*ArcT anh[(a*x)/Sqrt[1 + a^2*x^2]] - 2*(1 + a^2*x^2)^(3/2)*Cos[2*ArcTan[a*x]] - 3*a*x*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 51*a^3*x^3*ArcTan[a*x]*Log[ 1 - E^(I*ArcTan[a*x])] - 24*a^3*x^3*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a* x])] + 24*a^3*x^3*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 3*a*x*ArcTa n[a*x]*Log[1 + E^(I*ArcTan[a*x])] + 51*a^3*x^3*ArcTan[a*x]*Log[1 + E^(I*Ar cTan[a*x])] - (52*I)*a^3*x^3*PolyLog[2, -E^(I*ArcTan[a*x])] - (60*I)*a^3*x ^3*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (60*I)*a^3*x^3*ArcTan[ a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + (52*I)*a^3*x^3*PolyLog[2, E^(I*ArcT an[a*x])] + 60*a^3*x^3*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 60*a^3*x^3*Pol yLog[3, I*E^(I*ArcTan[a*x])] + 2*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Sin[2*Arc Tan[a*x]] + (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])]*Sin [3*ArcTan[a*x]] - (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x] )]*Sin[3*ArcTan[a*x]]))/(x^3*Sqrt[c + a^2*c*x^2])
Time = 10.47 (sec) , antiderivative size = 1119, normalized size of antiderivative = 1.66, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.208, Rules used = {5485, 5485, 5415, 224, 219, 5425, 5423, 3042, 4669, 3011, 2720, 5479, 5481, 242, 5485, 5425, 5423, 3042, 4669, 3011, 2720, 5479, 5493, 5489, 5497, 242, 5493, 5489, 7143}
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.
| \(\displaystyle \int \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}}{x^4} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{x^2}dx+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{x^4}dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \left (a^2 c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^4}dx\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \left (\frac {2}{3} a \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^3}dx-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 5481 |
| \(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \left (\frac {2}{3} a \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx+a c \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+c \left (\frac {2}{3} a \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \left (a^2 c \left (a^2 c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+c \left (\frac {2}{3} a \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \left (a^2 c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (a^2 c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+c \left (\frac {2}{3} a \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle c \left (a^2 c \left (\frac {a c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+c \left (\frac {2}{3} a \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \left (\frac {a c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (a^2 c \left (c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{\sqrt {a^2 c x^2+c}}\right )+c \left (\frac {2}{3} a \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \left (c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+c \left (c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (c \left (\frac {2}{3} a \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{\sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+c \left (c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \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)^2\right )}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (c \left (\frac {2}{3} a \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )-\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \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)^2\right )}{\sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c \left (c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right ) a^2+c \left (c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {2}{3} a \left (-\frac {\sqrt {a^2 c x^2+c} a}{x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx\right )-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 c x^3}\right )\right )\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle c \left (c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right ) a^2+c \left (c \left (2 a \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (2 a \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {2}{3} a \left (-\frac {\sqrt {a^2 c x^2+c} a}{x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx\right )-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 c x^3}\right )\right )\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle c \left (c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right ) a^2+c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {2}{3} a \left (-\frac {\sqrt {a^2 c x^2+c} a}{x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx\right )-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 c x^3}\right )\right )\) |
| \(\Big \downarrow \) 5489 |
| \(\displaystyle c \left (c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right ) a^2+c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {2}{3} a \left (-\frac {\sqrt {a^2 c x^2+c} a}{x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx\right )-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 c x^3}\right )\right )\) |
| \(\Big \downarrow \) 5497 |
| \(\displaystyle c \left (c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right ) a^2+c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {2}{3} a \left (-\frac {\sqrt {a^2 c x^2+c} a}{x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}-c \left (-\frac {1}{2} \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx a^2+\frac {1}{2} \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{2 c x^2}\right )\right )-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 c x^3}\right )\right )\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle c \left (c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right ) a^2+c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {2}{3} a \left (-\frac {\sqrt {a^2 c x^2+c} a}{x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}-c \left (-\frac {1}{2} \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx a^2-\frac {\sqrt {a^2 c x^2+c} a}{2 c x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{2 c x^2}\right )\right )-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 c x^3}\right )\right )\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle c \left (c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right ) a^2+c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {2}{3} a \left (-\frac {\sqrt {a^2 c x^2+c} a}{x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}-c \left (-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx a^2}{2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} a}{2 c x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{2 c x^2}\right )\right )-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 c x^3}\right )\right )\) |
| \(\Big \downarrow \) 5489 |
| \(\displaystyle c \left (c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right ) a^2+c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {2}{3} a \left (-\frac {\sqrt {a^2 c x^2+c} a}{x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}-c \left (-\frac {\sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right ) a^2}{2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} a}{2 c x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{2 c x^2}\right )\right )-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 c x^3}\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle c \left (c \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}+\frac {c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )}{2 a \sqrt {a^2 c x^2+c}}\right ) a^2+c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )+\frac {a c \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {2}{3} a \left (-\frac {\sqrt {a^2 c x^2+c} a}{x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}-c \left (-\frac {\sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right ) a^2}{2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} a}{2 c x}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{2 c x^2}\right )\right )-\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 c x^3}\right )\right )\) |
a^2*c*(a^2*c*(-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/a) + (x*Sqrt[c + a^2*c*x ^2]*ArcTan[a*x]^2)/2 + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] )/a + (c*Sqrt[1 + a^2*x^2]*((-2*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, (-I)*E ^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - Po lyLog[3, I*E^(I*ArcTan[a*x])])))/(2*a*Sqrt[c + a^2*c*x^2])) + c*(c*(-((Sqr t[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x)) + (2*a*Sqrt[1 + a^2*x^2]*(-2*ArcTan [a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 + I *a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/ Sqrt[c + a^2*c*x^2]) + (a*c*Sqrt[1 + a^2*x^2]*((-2*I)*ArcTan[E^(I*ArcTan[a *x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, I*E^(I *ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])])))/Sqrt[c + a^2*c*x^2])) + c*(c*(-1/3*((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(c*x^3) + (2*a*(-((a*Sq rt[c + a^2*c*x^2])/x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^2 - c*(-1/2*(a *Sqrt[c + a^2*c*x^2])/(c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*c*x^2) - (a^2*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/(2*Sqrt[c + a^2*c*x^2]))))/3) + a^2*c*( c*(-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x)) + (2*a*Sqrt[1 + a^2*x^2...
3.4.30.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2 *q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)*( a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
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_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1))) Int[(f*x) ^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x ])/(f*(m + 2))), x] + (Simp[d/(m + 2) Int[(f*x)^m*((a + b*ArcTan[c*x])/Sq rt[d + e*x^2]), x], x] - Simp[b*c*(d/(f*(m + 2))) Int[(f*x)^(m + 1)/Sqrt[ d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]
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_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_ Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sq rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c *x]], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 ]), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan [c*x])^p/(x*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_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Ar cTan[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1))) Int[(f*x)^(m + 1)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[c^2*((m + 2)/(f^2*(m + 1))) 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] && LtQ[m, -1] && NeQ[m, -2]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 4.44 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.59
| method | result | size |
| default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 a^{4} \arctan \left (a x \right )^{2} x^{4}-6 \arctan \left (a x \right ) x^{3} a^{3}-14 x^{2} \arctan \left (a x \right )^{2} a^{2}-2 a^{2} x^{2}-2 x \arctan \left (a x \right ) a -2 \arctan \left (a x \right )^{2}\right )}{6 x^{3}}+\frac {i c^{2} a^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+26 i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+30 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-30 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+30 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-30 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-12 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+26 \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+26 \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )\right )}{6 \sqrt {a^{2} x^{2}+1}}\) | \(401\) |
1/6*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(3*a^4*arctan(a*x)^2*x^4-6*arctan(a*x)*x ^3*a^3-14*x^2*arctan(a*x)^2*a^2-2*a^2*x^2-2*x*arctan(a*x)*a-2*arctan(a*x)^ 2)/x^3+1/6*I*c^2*a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(15*I*arctan(a*x)^2*ln(1+I* (1+I*a*x)/(a^2*x^2+1)^(1/2))-15*I*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+ 1)^(1/2))+26*I*arctan(a*x)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+30*arctan(a*x )*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-30*arctan(a*x)*polylog(2,I*(1+ I*a*x)/(a^2*x^2+1)^(1/2))+30*I*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3 0*I*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*arctan((1+I*a*x)/(a^2*x^2+ 1)^(1/2))+26*dilog((1+I*a*x)/(a^2*x^2+1)^(1/2))+26*dilog((1+I*a*x)/(a^2*x^ 2+1)^(1/2)+1))/(a^2*x^2+1)^(1/2)
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{4}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{4}} \,d x } \]
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^4} \,d x \]