\[ \int \coth ^{-1}\left (\sqrt {x}\right ) \, dx \]
Optimal antiderivative \[ x \,\mathrm {arccoth}\left (\sqrt {x}\right )-\arctanh \left (\sqrt {x}\right )+\sqrt {x} \]
command
integrate(arccoth(x^(1/2)),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
\[ \int \operatorname {arcoth}\left (\sqrt {x}\right )\,{d x} \]________________________________________________________________________________________