\[ \int \frac {\coth ^{-1}(a x)}{x^5} \, dx \]
Optimal antiderivative \[ -\frac {a}{12 x^{3}}-\frac {a^{3}}{4 x}-\frac {\mathrm {arccoth}\left (a x \right )}{4 x^{4}}+\frac {a^{4} \arctanh \left (a x \right )}{4} \]
command
integrate(arccoth(a*x)/x^5,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{3} \, a {\left (\frac {\frac {3 \, {\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} + \frac {3 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + 2 \, a^{3}}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{3}} + \frac {3 \, {\left (\frac {{\left (a x + 1\right )}^{3} a^{3}}{{\left (a x - 1\right )}^{3}} + \frac {{\left (a x + 1\right )} a^{3}}{a x - 1}\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{4}}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\operatorname {arcoth}\left (a x\right )}{x^{5}}\,{d x} \]________________________________________________________________________________________