\[ \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))} \, dx \]
Optimal antiderivative \[ -\frac {\ln \left (x \right )}{b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {\ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )} \]
command
integrate(1/x/arccoth(tanh(b*x+a)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, \log \left (\pi - 2 i \, b x - 2 i \, a\right )}{-i \, \pi - 2 \, a} - \frac {2 i \, \log \left (x\right )}{\pi - 2 i \, a} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{x \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]________________________________________________________________________________________