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