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