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