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