\[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^2} \, dx \]
Optimal antiderivative \[ -\frac {3 b^{2}}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{3} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {3 b}{2 x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {1}{2 x^{2} \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right ) \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {3 b^{2} \ln \left (x \right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{4}}-\frac {3 b^{2} \ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{4}} \]
command
integrate(1/x^3/arccoth(tanh(b*x+a))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {48 \, b^{2} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}} + \frac {48 \, b^{2} \log \left (x\right )}{\pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}} + \frac {16 \, b^{2}}{-2 i \, \pi ^{3} b x - 12 \, \pi ^{2} a b x + 24 i \, \pi a^{2} b x + 16 \, a^{3} b x + \pi ^{4} - 8 i \, \pi ^{3} a - 24 \, \pi ^{2} a^{2} + 32 i \, \pi a^{3} + 16 \, a^{4}} - \frac {4 \, {\left (i \, \pi - 8 \, b x + 2 \, a\right )}}{-2 i \, \pi ^{3} x^{2} - 12 \, \pi ^{2} a x^{2} + 24 i \, \pi a^{2} x^{2} + 16 \, a^{3} x^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]________________________________________________________________________________________