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