\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{a \,c^{2}}-\frac {4}{3 a \,c^{2} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}} \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^{2} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}} \sqrt {1+\frac {1}{a x}}}-\frac {11}{3 a \,c^{2} \sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a \,c^{2} \sqrt {1+\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^2,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
\[ \frac {1}{12} \, a {\left (\frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {12 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{2}} - \frac {{\left (a x + 1\right )} {\left (\frac {18 \, {\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} + \frac {3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}} - \frac {24 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]