\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {4}{3 a \,c^{2} \left (c -\frac {c}{a x}\right )^{\frac {3}{2}}}+\frac {x}{c^{2} \left (c -\frac {c}{a x}\right )^{\frac {3}{2}}}+\frac {3 \arctanh \! \left (\frac {\sqrt {c -\frac {c}{a x}}}{\sqrt {c}}\right )}{a \,c^{\frac {7}{2}}}+\frac {\arctanh \! \left (\frac {\sqrt {c -\frac {c}{a x}}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{4 a \,c^{\frac {7}{2}}}-\frac {7}{2 a \,c^{3} \sqrt {c -\frac {c}{a x}}} \]
command
integrate((a*x-1)/(a*x+1)/(c-c/a/x)^(7/2),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
\[ -\frac {1}{12} \, a c {\left (\frac {2 \, {\left (2 \, c + \frac {15 \, {\left (a c x - c\right )}}{a x}\right )} x}{{\left (a c x - c\right )} a c^{4} \sqrt {\frac {a c x - c}{a x}}} + \frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a c x - c}{a x}}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{4}} + \frac {36 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{4}} - \frac {12 \, \sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )} c^{4}}\right )} \]