\[ \int \frac {x \coth ^{-1}(x)}{\left (1-x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {x}{4 \left (-x^{2}+1\right )}+\frac {\mathrm {arccoth}\left (x \right )}{-2 x^{2}+2}-\frac {\arctanh \left (x \right )}{4} \]
command
integrate(x*arccoth(x)/(-x^2+1)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{16} \, {\left (\frac {x + 1}{x - 1} + \frac {x - 1}{x + 1}\right )} \log \left (-\frac {\frac {\frac {x + 1}{x - 1} - 1}{\frac {x + 1}{x - 1} + 1} + 1}{\frac {\frac {x + 1}{x - 1} - 1}{\frac {x + 1}{x - 1} + 1} - 1}\right ) + \frac {x + 1}{16 \, {\left (x - 1\right )}} - \frac {x - 1}{16 \, {\left (x + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x \operatorname {arcoth}\left (x\right )}{{\left (x^{2} - 1\right )}^{2}}\,{d x} \]________________________________________________________________________________________