\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x} \, dx \]
Optimal antiderivative \[ -b x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )+\frac {\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}{2}+\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2} \ln \left (x \right ) \]
command
integrate(arccoth(tanh(b*x+a))^2/x,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2} \, b^{2} x^{2} + {\left (i \, \pi b + 2 \, a b\right )} x - \frac {1}{4} \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (x\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{x}\,{d x} \]________________________________________________________________________________________