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