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