\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx \]
Optimal antiderivative \[ -\frac {8}{7 a \,c^{4} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}}}-\frac {11}{7 a \,c^{4} \left (1-\frac {1}{a x}\right )^{\frac {5}{2}} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}}}-\frac {62}{21 a \,c^{4} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}}}+\frac {x}{c^{4} \left (1-\frac {1}{a x}\right )^{\frac {7}{2}} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}}}+\frac {\arctanh \! \left (\sqrt {1-\frac {1}{a x}}\, \sqrt {1+\frac {1}{a x}}\right )}{a \,c^{4}}-\frac {269}{21 a \,c^{4} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}} \sqrt {1-\frac {1}{a x}}}+\frac {262 \sqrt {1-\frac {1}{a x}}}{35 a \,c^{4} \left (1+\frac {1}{a x}\right )^{\frac {5}{2}}}+\frac {163 \sqrt {1-\frac {1}{a x}}}{35 a \,c^{4} \left (1+\frac {1}{a x}\right )^{\frac {3}{2}}}+\frac {128 \sqrt {1-\frac {1}{a x}}}{35 a \,c^{4} \sqrt {1+\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^4,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
\[ \frac {1}{6720} \, a {\left (\frac {6720 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {6720 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{4}} - \frac {5 \, {\left (a x + 1\right )}^{3} {\left (\frac {42 \, {\left (a x - 1\right )}}{a x + 1} + \frac {329 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2940 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{4} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {13440 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} + \frac {7 \, {\left (\frac {50 \, {\left (a x - 1\right )} a^{8} c^{16} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2} a^{8} c^{16} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 705 \, a^{8} c^{16} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{10} c^{20}}\right )} \]