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