\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx \]
Optimal antiderivative \[ \frac {16}{35 \sqrt {\frac {a x -1}{a x +1}}\, a \,c^{4}}-\frac {-6 a x +1}{35 \sqrt {\frac {a x -1}{a x +1}}\, a \,c^{4} \left (-a^{2} x^{2}+1\right )^{3}}-\frac {2 \left (-4 a x +1\right )}{35 \sqrt {\frac {a x -1}{a x +1}}\, a \,c^{4} \left (-a^{2} x^{2}+1\right )^{2}}-\frac {8 \left (-2 a x +1\right )}{35 \sqrt {\frac {a x -1}{a x +1}}\, a \,c^{4} \left (-a^{2} x^{2}+1\right )} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(-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
\[ \frac {\frac {{\left (a x + 1\right )}^{3} {\left (\frac {42 \, {\left (a x - 1\right )}}{a x + 1} - \frac {175 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {700 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5\right )}}{{\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {70 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {7 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 525 \, \sqrt {\frac {a x - 1}{a x + 1}}}{2240 \, a c^{4}} \]