\[ \int \frac {x \coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {x}{16 \left (-x^{2}+1\right )^{2}}-\frac {3 x}{32 \left (-x^{2}+1\right )}+\frac {\mathrm {arccoth}\left (x \right )}{4 \left (-x^{2}+1\right )^{2}}-\frac {3 \arctanh \left (x \right )}{32} \]
command
integrate(x*arccoth(x)/(-x^2+1)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{128} \, {\left (\frac {{\left (x - 1\right )}^{2} {\left (\frac {4 \, {\left (x + 1\right )}}{x - 1} - 1\right )}}{{\left (x + 1\right )}^{2}} - \frac {{\left (x + 1\right )}^{2}}{{\left (x - 1\right )}^{2}} + \frac {4 \, {\left (x + 1\right )}}{x - 1}\right )} \log \left (-\frac {\frac {\frac {x + 1}{x - 1} - 1}{\frac {x + 1}{x - 1} + 1} + 1}{\frac {\frac {x + 1}{x - 1} - 1}{\frac {x + 1}{x - 1} + 1} - 1}\right ) - \frac {{\left (x - 1\right )}^{2} {\left (\frac {8 \, {\left (x + 1\right )}}{x - 1} - 1\right )}}{256 \, {\left (x + 1\right )}^{2}} - \frac {{\left (x + 1\right )}^{2}}{256 \, {\left (x - 1\right )}^{2}} + \frac {x + 1}{32 \, {\left (x - 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int -\frac {x \operatorname {arcoth}\left (x\right )}{{\left (x^{2} - 1\right )}^{3}}\,{d x} \]________________________________________________________________________________________