\[ \int x \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {c \left (a +b \,\mathrm {arccoth}\left (\frac {x^{2}}{c}\right )\right )^{2}}{2}+\frac {x^{2} \left (a +b \,\mathrm {arccoth}\left (\frac {x^{2}}{c}\right )\right )^{2}}{2}-b c \left (a +b \,\mathrm {arccoth}\left (\frac {x^{2}}{c}\right )\right ) \ln \left (2-\frac {2}{1+\frac {c}{x^{2}}}\right )+\frac {b^{2} c \polylog \left (2, -1+\frac {2}{1+\frac {c}{x^{2}}}\right )}{2} \]
command
int(x*(a+b*arctanh(c/x^2))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(6869\) |
Maple 2021.1 output
\[ \int x \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}\, dx \]________________________________________________________________________________________