\[ \int \frac {\cosh ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {a \ln \left (x \right ) \sqrt {a x -1}}{\sqrt {-a x +1}}-\frac {\mathrm {arccosh}\left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{x} \]
command
integrate(arccosh(a*x)/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {a x \arctan \left (\frac {\sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} {\left (x^{2} + 1\right )}}{a^{2} x^{4} - {\left (a^{2} + 1\right )} x^{2} + 1}\right ) - \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]