\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {4 \left (a +\frac {1}{x}\right )}{3 a^{2} c^{2} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {3}{2}}}+\frac {3 \arctanh \! \left (\sqrt {1-\frac {1}{a^{2} x^{2}}}\right )}{a \,c^{2}}+\frac {-9 a -\frac {11}{x}}{3 a^{2} c^{2} \sqrt {1-\frac {1}{a^{2} x^{2}}}}+\frac {x \sqrt {1-\frac {1}{a^{2} x^{2}}}}{c^{2}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^2,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
\[ \frac {1}{3} \, a {\left (\frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {9 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{2}} - \frac {{\left (a x + 1\right )} {\left (\frac {12 \, {\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]