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