\[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))} \, dx \]
Optimal antiderivative \[ \frac {b}{x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2}}+\frac {1}{2 x^{2} \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}-\frac {b^{2} \ln \left (x \right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{3}}+\frac {b^{2} \ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{3}} \]
command
integrate(1/x^3/arccoth(tanh(b*x+a)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {8 \, b^{2} \log \left (\pi - 2 i \, b x - 2 i \, a\right )}{-i \, \pi ^{3} - 6 \, \pi ^{2} a + 12 i \, \pi a^{2} + 8 \, a^{3}} + \frac {8 i \, b^{2} \log \left (x\right )}{\pi ^{3} - 6 i \, \pi ^{2} a - 12 \, \pi a^{2} + 8 i \, a^{3}} - \frac {-i \, \pi + 4 \, b x - 2 \, a}{\pi ^{2} x^{2} - 4 i \, \pi a x^{2} - 4 \, a^{2} x^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]________________________________________________________________________________________