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