Integrand size = 24, antiderivative size = 254 \[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\frac {x \sqrt {1-a^2 x^2}}{12 a^2}-\frac {\arcsin (a x)}{6 a^3}+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{12 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2+\frac {\arctan \left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2}{4 a^3}-\frac {i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )}{4 a^3}+\frac {i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )}{4 a^3}+\frac {i \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )}{4 a^3}-\frac {i \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )}{4 a^3} \]
-1/6*arcsin(a*x)/a^3+1/4*arctan((a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)^2 /a^3-1/4*I*arctanh(a*x)*polylog(2,-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^3+1/4*I *arctanh(a*x)*polylog(2,I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^3+1/4*I*polylog(3, -I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^3-1/4*I*polylog(3,I*(a*x+1)/(-a^2*x^2+1)^ (1/2))/a^3+1/12*x*(-a^2*x^2+1)^(1/2)/a^2+1/12*arctanh(a*x)*(-a^2*x^2+1)^(1 /2)/a^3+1/6*x^2*arctanh(a*x)*(-a^2*x^2+1)^(1/2)/a-1/8*x*arctanh(a*x)^2*(-a ^2*x^2+1)^(1/2)/a^2+1/4*x^3*arctanh(a*x)^2*(-a^2*x^2+1)^(1/2)
Time = 0.90 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.90 \[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\frac {\sqrt {1-a^2 x^2} \left (6 \text {arctanh}(a x)-4 \left (1-a^2 x^2\right ) \text {arctanh}(a x)-6 a x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+a x \left (2+3 \text {arctanh}(a x)^2\right )-\frac {i \left (-8 i \arctan \left (\tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+3 \text {arctanh}(a x)^2 \log \left (1-i e^{-\text {arctanh}(a x)}\right )-3 \text {arctanh}(a x)^2 \log \left (1+i e^{-\text {arctanh}(a x)}\right )+6 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-6 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )+6 \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )-6 \operatorname {PolyLog}\left (3,i e^{-\text {arctanh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{24 a^3} \]
(Sqrt[1 - a^2*x^2]*(6*ArcTanh[a*x] - 4*(1 - a^2*x^2)*ArcTanh[a*x] - 6*a*x* (1 - a^2*x^2)*ArcTanh[a*x]^2 + a*x*(2 + 3*ArcTanh[a*x]^2) - (I*((-8*I)*Arc Tan[Tanh[ArcTanh[a*x]/2]] + 3*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - 3 *ArcTanh[a*x]^2*Log[1 + I/E^ArcTanh[a*x]] + 6*ArcTanh[a*x]*PolyLog[2, (-I) /E^ArcTanh[a*x]] - 6*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] + 6*PolyLog [3, (-I)/E^ArcTanh[a*x]] - 6*PolyLog[3, I/E^ArcTanh[a*x]]))/Sqrt[1 - a^2*x ^2]))/(24*a^3)
Time = 4.26 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.87, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6576, 6578, 6514, 3042, 4668, 3011, 2720, 6556, 223, 6578, 262, 223, 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.
\(\displaystyle \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx\) |
\(\Big \downarrow \) 6576 |
\(\displaystyle \int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx-a^2 \int \frac {x^4 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 6578 |
\(\displaystyle \frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {\int \frac {\text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\left (a^2 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 6514 |
\(\displaystyle \frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\left (a^2 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2d\text {arctanh}(a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \text {arctanh}(a x)^2 \csc \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )d\text {arctanh}(a x)}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\left (a^2 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {-2 i \int \text {arctanh}(a x) \log \left (1-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \int \text {arctanh}(a x) \log \left (1+i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\left (a^2 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {2 i \left (\int \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\left (a^2 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\left (a^2 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 6556 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\left (a^2 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {3 \int \frac {x^2 \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int \frac {x^3 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 6578 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2}}dx}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}}{2 a}+\frac {3 \left (\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {\int \frac {\text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}}{2 a}+\frac {3 \left (\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {\int \frac {\text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {3 \left (\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {\int \frac {\text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}+\frac {\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 6514 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}+\frac {3 \left (\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {\int \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2d\text {arctanh}(a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}+\frac {3 \left (\frac {\int \text {arctanh}(a x)^2 \csc \left (i \text {arctanh}(a x)+\frac {\pi }{2}\right )d\text {arctanh}(a x)}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}+\frac {3 \left (\frac {-2 i \int \text {arctanh}(a x) \log \left (1-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \int \text {arctanh}(a x) \log \left (1+i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}+\frac {3 \left (\frac {2 i \left (\int \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {\frac {2 \int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}+\frac {3 \left (\frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\int \frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 6556 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {\frac {2 \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}+\frac {3 \left (\frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}-\left (a^2 \left (\frac {3 \left (\frac {2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{2 a^3}+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}+\frac {\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}\right )\right )+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )}{2 a^3}+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}-\left (a^2 \left (-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}+\frac {3 \left (\frac {2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )}{2 a^3}+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}+\frac {\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}\right )\right )\) |
-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^ArcTanh[a*x]]))/(2*a^3) - a^2*(-1/4*(x^3*Sqrt[1 - a^2*x^2 ]*ArcTanh[a*x]^2)/a^2 + ((-1/2*(x*Sqrt[1 - a^2*x^2])/a^2 + ArcSin[a*x]/(2* a^3))/(3*a) - (x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(3*a^2) + (2*(ArcSin[a* x]/a^2 - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/a^2))/(3*a^2))/(2*a) + (3*(-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]]) + Pol yLog[3, I*E^ArcTanh[a*x]]))/(2*a^3)))/(4*a^2))
3.5.40.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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_)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x ^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ [p, 1] && IntegerQ[q]))
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]
\[\int x^{2} \operatorname {arctanh}\left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}d x\]
\[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int { \sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \]
\[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \]
\[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int { \sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \]
\[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int { \sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \]
Timed out. \[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int x^2\,{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2} \,d x \]