\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx \]
Optimal antiderivative \[ -\frac {16 \left (a +\frac {1}{x}\right )}{7 a^{2} c^{4} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {7}{2}}}-\frac {4 \left (7 a +\frac {17}{x}\right )}{35 a^{2} c^{4} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {5}{2}}}+\frac {-175 a -\frac {307}{x}}{105 a^{2} c^{4} \left (1-\frac {1}{a^{2} x^{2}}\right )^{\frac {3}{2}}}+\frac {5 \arctanh \! \left (\sqrt {1-\frac {1}{a^{2} x^{2}}}\right )}{a \,c^{4}}+\frac {-525 a -\frac {719}{x}}{105 a^{2} c^{4} \sqrt {1-\frac {1}{a^{2} x^{2}}}}+\frac {x \sqrt {1-\frac {1}{a^{2} x^{2}}}}{c^{4}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^4,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
\[ \frac {1}{420} \, a {\left (\frac {2100 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {2100 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{4}} - \frac {{\left (a x + 1\right )}^{3} {\left (\frac {126 \, {\left (a x - 1\right )}}{a x + 1} + \frac {595 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {3360 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 15\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{4} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {840 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]