\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {32 \left (a +\frac {1}{x}\right )}{7 a^{2} c^{3} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {7}{2}}}-\frac {2 \left (7 a +\frac {13}{x}\right )}{7 a^{2} c^{3} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {3}{2}}}-\frac {16}{7 a^{2} c^{3} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {5}{2}} x}+\frac {6 \arctanh \! \left (\sqrt {1-\frac {1}{a^{2} x^{2}}}\right )}{a \,c^{3}}+\frac {-42 a -\frac {59}{x}}{7 a^{2} c^{3} \sqrt {1-\frac {1}{a^{2} x^{2}}}}+\frac {x \sqrt {1-\frac {1}{a^{2} x^{2}}}}{c^{3}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {1}{14} \, a {\left (\frac {84 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {84 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{3}} - \frac {{\left (a x + 1\right )}^{3} {\left (\frac {7 \, {\left (a x - 1\right )}}{a x + 1} + \frac {28 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {140 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {28 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]