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