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