\(\int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx\) [440]

Optimal result

Integrand size = 24, antiderivative size = 254 \[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\frac {x \sqrt {1-a^2 x^2}}{12 a^2}-\frac {\arcsin (a x)}{6 a^3}+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{12 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2+\frac {\arctan \left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2}{4 a^3}-\frac {i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )}{4 a^3}+\frac {i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )}{4 a^3}+\frac {i \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )}{4 a^3}-\frac {i \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )}{4 a^3} \] Output:

1/12*x*(-a^2*x^2+1)^(1/2)/a^2-1/6*arcsin(a*x)/a^3+1/12*(-a^2*x^2+1)^(1/2)* 
arctanh(a*x)/a^3+1/6*x^2*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a-1/8*x*(-a^2*x^2 
+1)^(1/2)*arctanh(a*x)^2/a^2+1/4*x^3*(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2+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
 

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.90 \[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\frac {\sqrt {1-a^2 x^2} \left (6 \text {arctanh}(a x)-4 \left (1-a^2 x^2\right ) \text {arctanh}(a x)-6 a x \left (1-a^2 x^2\right ) \text {arctanh}(a x)^2+a x \left (2+3 \text {arctanh}(a x)^2\right )-\frac {i \left (-8 i \arctan \left (\tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+3 \text {arctanh}(a x)^2 \log \left (1-i e^{-\text {arctanh}(a x)}\right )-3 \text {arctanh}(a x)^2 \log \left (1+i e^{-\text {arctanh}(a x)}\right )+6 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-6 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )+6 \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )-6 \operatorname {PolyLog}\left (3,i e^{-\text {arctanh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{24 a^3} \] Input:

Integrate[x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]
 

Output:

(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)
 

Rubi [A] (verified)

Time = 4.30 (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

\(\displaystyle \frac {2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )}{2 a^3}+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}-\left (a^2 \left (-\frac {x^3 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{4 a^2}+\frac {3 \left (\frac {2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )+2 i \left (\operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )\right )-2 i \left (\operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )\right )}{2 a^3}+\frac {\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}}{a}-\frac {x \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 a^2}\right )}{4 a^2}+\frac {\frac {2 \left (\frac {\arcsin (a x)}{a^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a^2}\right )}{3 a^2}-\frac {x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{3 a^2}+\frac {\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}}{3 a}}{2 a}\right )\right )\)

Input:

Int[x^2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2,x]
 

Output:

-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))
 

Defintions of rubi rules used

rule 223

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]
 

rule 262

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]
 

rule 2720

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]]]
 

rule 3011

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4668

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]
 

rule 6514

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]
 

rule 6556

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]
 

rule 6576

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]))
 

rule 6578

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]
 

rule 7143

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]
 

Maple [F]

\[\int x^{2} \sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )^{2}d x\]

Input:

int(x^2*(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2,x)
 

Output:

int(x^2*(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2,x)
 

Fricas [F]

\[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int { \sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \] Input:

integrate(x^2*(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2,x, algorithm="fricas")
 

Output:

integral(sqrt(-a^2*x^2 + 1)*x^2*arctanh(a*x)^2, x)
 

Sympy [F]

\[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \] Input:

integrate(x**2*(-a**2*x**2+1)**(1/2)*atanh(a*x)**2,x)
 

Output:

Integral(x**2*sqrt(-(a*x - 1)*(a*x + 1))*atanh(a*x)**2, x)
 

Maxima [F]

\[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int { \sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \] Input:

integrate(x^2*(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2,x, algorithm="maxima")
 

Output:

integrate(sqrt(-a^2*x^2 + 1)*x^2*arctanh(a*x)^2, x)
 

Giac [F]

\[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int { \sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{2} \,d x } \] Input:

integrate(x^2*(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2,x, algorithm="giac")
 

Output:

integrate(sqrt(-a^2*x^2 + 1)*x^2*arctanh(a*x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int x^2\,{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2} \,d x \] Input:

int(x^2*atanh(a*x)^2*(1 - a^2*x^2)^(1/2),x)
 

Output:

int(x^2*atanh(a*x)^2*(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int x^2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2 \, dx=\int \sqrt {-a^{2} x^{2}+1}\, \mathit {atanh} \left (a x \right )^{2} x^{2}d x \] Input:

int(x^2*(-a^2*x^2+1)^(1/2)*atanh(a*x)^2,x)
 

Output:

int(sqrt( - a**2*x**2 + 1)*atanh(a*x)**2*x**2,x)