38.4 Problem number 20

\[ \int e^{3 \coth ^{-1}(a x)} \, dx \]

Optimal antiderivative \[ \frac {3 \arctanh \! \left (\sqrt {1-\frac {1}{a^{2} x^{2}}}\right )}{a}-\frac {4 \sqrt {1-\frac {1}{a^{2} x^{2}}}}{a -\frac {1}{x}}+x \sqrt {1-\frac {1}{a^{2} x^{2}}} \]

command

integrate(1/((a*x-1)/(a*x+1))^(3/2),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

\[ a {\left (\frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (\frac {3 \, {\left (a x - 1\right )}}{a x + 1} - 2\right )}}{a^{2} {\left (\frac {{\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - \sqrt {\frac {a x - 1}{a x + 1}}\right )}}\right )} \]