\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {3 \arctanh \! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{a c}-\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{\frac {3}{2}}}+\frac {x \sqrt {1+\frac {1}{a x}}}{c \left (1-\frac {1}{a x}\right )^{\frac {3}{2}}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/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
\[ \frac {1}{3} \, a {\left (\frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {9 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c} - \frac {{\left (a x + 1\right )} {\left (\frac {12 \, {\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]