\[ \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {1}{6 x^{\frac {3}{2}}}-\frac {\mathrm {arccoth}\left (\sqrt {x}\right )}{2 x^{2}}+\frac {\arctanh \left (\sqrt {x}\right )}{2}-\frac {1}{2 \sqrt {x}} \]
command
integrate(arccoth(x^(1/2))/x^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (\frac {3 \, {\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + \frac {3 \, {\left (\sqrt {x} + 1\right )}}{\sqrt {x} - 1} + 2\right )}}{3 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1\right )}^{3}} + \frac {2 \, {\left (\frac {{\left (\sqrt {x} + 1\right )}^{3}}{{\left (\sqrt {x} - 1\right )}^{3}} + \frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )} \log \left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{{\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1\right )}^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{x^{3}}\,{d x} \]________________________________________________________________________________________