Integrand size = 24, antiderivative size = 219 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\arcsin (a x)}{a^4}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^4}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^3}+\frac {5 \arctan \left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2}{a^4}-\frac {2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}-\frac {5 i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )}{a^4}+\frac {5 i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )}{a^4}+\frac {5 i \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )}{a^4}-\frac {5 i \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )}{a^4} \]
arcsin(a*x)/a^4+5*arctan((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^2/a^4-5* I*arctanh(a*x)*polylog(2,-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^4+5*I*arctanh(a* x)*polylog(2,I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^4+5*I*polylog(3,-I*(a*x+1)/(- a^2*x^2+1)^(1/2))/a^4-5*I*polylog(3,I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^4-arct anh(a*x)*(-a^2*x^2+1)^(1/2)/a^4-1/2*x*arctanh(a*x)^2*(-a^2*x^2+1)^(1/2)/a^ 3-2/3*arctanh(a*x)^3*(-a^2*x^2+1)^(1/2)/a^4-1/3*x^2*arctanh(a*x)^3*(-a^2*x ^2+1)^(1/2)/a^2
Time = 0.70 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-3 a x \text {arctanh}(a x)^2+2 \left (1-a^2 x^2\right ) \text {arctanh}(a x)^3-6 \text {arctanh}(a x) \left (1+\text {arctanh}(a x)^2\right )-\frac {3 i \left (4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+5 \text {arctanh}(a x)^2 \log \left (1-i e^{-\text {arctanh}(a x)}\right )-5 \text {arctanh}(a x)^2 \log \left (1+i e^{-\text {arctanh}(a x)}\right )+10 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-10 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )+10 \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )-10 \operatorname {PolyLog}\left (3,i e^{-\text {arctanh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{6 a^4} \]
(Sqrt[1 - a^2*x^2]*(-3*a*x*ArcTanh[a*x]^2 + 2*(1 - a^2*x^2)*ArcTanh[a*x]^3 - 6*ArcTanh[a*x]*(1 + ArcTanh[a*x]^2) - ((3*I)*((4*I)*ArcTan[Tanh[ArcTanh [a*x]/2]] + 5*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - 5*ArcTanh[a*x]^2* Log[1 + I/E^ArcTanh[a*x]] + 10*ArcTanh[a*x]*PolyLog[2, (-I)/E^ArcTanh[a*x] ] - 10*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 10*PolyLog[3, (-I)/E^Ar cTanh[a*x]] - 10*PolyLog[3, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x^2]))/(6*a^4 )
Time = 2.68 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.45, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6578, 6556, 6514, 3042, 4668, 3011, 2720, 6578, 6514, 3042, 4668, 3011, 2720, 6556, 223, 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.
| \(\displaystyle \int \frac {x^3 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 6578 |
| \(\displaystyle \frac {\int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {2 \int \frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle \frac {2 \left (\frac {3 \int \frac {\text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}\right )}{3 a^2}+\frac {\int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 6514 |
| \(\displaystyle \frac {\int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {2 \left (\frac {3 \int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2d\text {arctanh}(a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \int \text {arctanh}(a x)^2 \csc \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )d\text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (-2 i \int \text {arctanh}(a x) \log \left (1-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \int \text {arctanh}(a x) \log \left (1+i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 6578 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {\int \frac {\text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 6514 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {\int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2d\text {arctanh}(a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\frac {\int \text {arctanh}(a x)^2 \csc \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )d\text {arctanh}(a x)}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\frac {-2 i \int \text {arctanh}(a x) \log \left (1-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \int \text {arctanh}(a x) \log \left (1+i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\frac {2 i \left (\int \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 6556 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )\right )}{a^2}\right )}{3 a^2}+\frac {\frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}}{a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{a^2}+\frac {3 \left (2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )\right )}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{3 a^2}+\frac {\frac {2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )}{2 a^3}+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}}{a}\) |
-1/3*(x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/a^2 + (-1/2*(x*Sqrt[1 - a^2*x^ 2]*ArcTanh[a*x]^2)/a^2 + (ArcSin[a*x]/a^2 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x ])/a^2)/a + (2*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 + (2*I)*(-(ArcTanh[a* x]*PolyLog[2, (-I)*E^ArcTanh[a*x]]) + PolyLog[3, (-I)*E^ArcTanh[a*x]]) - ( 2*I)*(-(ArcTanh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]]) + PolyLog[3, I*E^ArcTan h[a*x]]))/(2*a^3))/a + (2*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/a^2) + (3* (2*ArcTan[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 + (2*I)*(-(ArcTanh[a*x]*PolyLog[2 , (-I)*E^ArcTanh[a*x]]) + PolyLog[3, (-I)*E^ArcTanh[a*x]]) - (2*I)*(-(ArcT anh[a*x]*PolyLog[2, I*E^ArcTanh[a*x]]) + PolyLog[3, I*E^ArcTanh[a*x]])))/a ^2))/(3*a^2)
3.4.80.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
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_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[1/(c*Sqrt[d]) Subst[Int[(a + b*x)^p*Sech[x], x], x, ArcTa nh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0 ] && GtQ[d, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q _.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q + 1))), x] + Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTanh[c* x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
Int[(((a_.) + ArcTanh[(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*ArcTanh[c*x])^p/(c^2*d*m)), x] + (Simp[b*f*(p/(c*m)) Int[(f*x)^(m - 1 )*((a + b*ArcTanh[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*ArcTanh[c*x])^p/Sqrt[d + e*x^2]), x] , x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && GtQ[m, 1]
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]
\[\int \frac {x^{3} \operatorname {arctanh}\left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}}d x\]
\[ \int \frac {x^3 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{3} \operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
\[ \int \frac {x^3 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{3} \operatorname {atanh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int \frac {x^3 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{3} \operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
Exception generated. \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3 \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^3\,{\mathrm {atanh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]