\[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} \, dx \]
Optimal antiderivative \[ \left (1-\frac {1}{a x}\right )^{\frac {7}{8}} \left (1+\frac {1}{a x}\right )^{\frac {1}{8}} x +\frac {\arctan \! \left (\frac {\left (1+\frac {1}{a x}\right )^{\frac {1}{8}}}{\left (1-\frac {1}{a x}\right )^{\frac {1}{8}}}\right )}{2 a}+\frac {\arctanh \! \left (\frac {\left (1+\frac {1}{a x}\right )^{\frac {1}{8}}}{\left (1-\frac {1}{a x}\right )^{\frac {1}{8}}}\right )}{2 a}-\frac {\arctan \! \left (1-\frac {\left (1+\frac {1}{a x}\right )^{\frac {1}{8}} \sqrt {2}}{\left (1-\frac {1}{a x}\right )^{\frac {1}{8}}}\right ) \sqrt {2}}{4 a}+\frac {\arctan \! \left (1+\frac {\left (1+\frac {1}{a x}\right )^{\frac {1}{8}} \sqrt {2}}{\left (1-\frac {1}{a x}\right )^{\frac {1}{8}}}\right ) \sqrt {2}}{4 a}-\frac {\ln \! \left (1+\frac {\left (1+\frac {1}{a x}\right )^{\frac {1}{4}}}{\left (1-\frac {1}{a x}\right )^{\frac {1}{4}}}-\frac {\left (1+\frac {1}{a x}\right )^{\frac {1}{8}} \sqrt {2}}{\left (1-\frac {1}{a x}\right )^{\frac {1}{8}}}\right ) \sqrt {2}}{8 a}+\frac {\ln \! \left (1+\frac {\left (1+\frac {1}{a x}\right )^{\frac {1}{4}}}{\left (1-\frac {1}{a x}\right )^{\frac {1}{4}}}+\frac {\left (1+\frac {1}{a x}\right )^{\frac {1}{8}} \sqrt {2}}{\left (1-\frac {1}{a x}\right )^{\frac {1}{8}}}\right ) \sqrt {2}}{8 a} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/8),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
\[ -\frac {1}{8} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a^{2}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}\right )}{a^{2}} - \frac {\sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{2}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{2}} + \frac {4 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}\right )}{a^{2}} - \frac {2 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + 1\right )}{a^{2}} + \frac {2 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} - 1 \right |}\right )}{a^{2}} + \frac {16 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]