\[ \int x \coth ^{-1}\left (\sqrt {x}\right ) \, dx \]
Optimal antiderivative \[ \frac {x^{\frac {3}{2}}}{6}+\frac {x^{2} \mathrm {arccoth}\left (\sqrt {x}\right )}{2}-\frac {\arctanh \left (\sqrt {x}\right )}{2}+\frac {\sqrt {x}}{2} \]
command
integrate(x*arccoth(x^(1/2)),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
\[ \int x \operatorname {arcoth}\left (\sqrt {x}\right )\,{d x} \]________________________________________________________________________________________