\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {8}{15 \sqrt {\frac {a x -1}{a x +1}}\, a \,c^{3}}-\frac {-4 a x +1}{15 \sqrt {\frac {a x -1}{a x +1}}\, a \,c^{3} \left (-a^{2} x^{2}+1\right )^{2}}-\frac {4 \left (-2 a x +1\right )}{15 \sqrt {\frac {a x -1}{a x +1}}\, a \,c^{3} \left (-a^{2} x^{2}+1\right )} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-a^2*c*x^2+c)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {\frac {{\left (a x + 1\right )}^{2} {\left (\frac {20 \, {\left (a x - 1\right )}}{a x + 1} - \frac {90 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3\right )}}{{\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}} + \frac {5 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - 60 \, \sqrt {\frac {a x - 1}{a x + 1}}}{240 \, a c^{3}} \]