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