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