\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx \]
Optimal antiderivative \[ \frac {3 \arctanh \! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{a \,c^{3}}-\frac {8}{7 a \,c^{3} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} \sqrt {1+\frac {1}{a x}}}-\frac {53}{35 a \,c^{3} \left (1-\frac {1}{a x}\right )^{\frac {5}{2}} \sqrt {1+\frac {1}{a x}}}-\frac {88}{35 a \,c^{3} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}} \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^{3} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} \sqrt {1+\frac {1}{a x}}}-\frac {281}{35 a \,c^{3} \sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a \,c^{3} \sqrt {1+\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^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
\[ \frac {1}{560} \, a {\left (\frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {1680 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{3}} - \frac {{\left (a x + 1\right )}^{3} {\left (\frac {56 \, {\left (a x - 1\right )}}{a x + 1} + \frac {350 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2520 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 5\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}} + \frac {35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} - \frac {1120 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]