\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx \]
Optimal antiderivative \[ -\frac {16}{63 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a \,c^{4}}-\frac {-2 a x +1}{9 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a \,c^{4} \left (-a^{2} x^{2}+1\right )^{3}}-\frac {10 \left (-4 a x +3\right )}{63 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a \,c^{4} \left (-a^{2} x^{2}+1\right )^{2}}+\frac {-\frac {16 a x}{21}+\frac {8}{7}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a \,c^{4} \left (-a^{2} x^{2}+1\right )} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(-a^2*c*x^2+c)^4,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
\[ \frac {\frac {{\left (a x + 1\right )}^{4} {\left (\frac {54 \, {\left (a x - 1\right )}}{a x + 1} - \frac {189 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {420 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {945 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - 7\right )}}{{\left (a x - 1\right )}^{4} \sqrt {\frac {a x - 1}{a x + 1}}} + \frac {21 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - 378 \, \sqrt {\frac {a x - 1}{a x + 1}}}{4032 \, a c^{4}} \]