\[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^2} \, dx \]
Optimal antiderivative \[ -\frac {2 b}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {1}{x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right ) \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {2 b \ln \left (x \right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{3}}-\frac {2 b \ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{3}} \]
command
integrate(1/x^2/arccoth(tanh(b*x+a))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {16 \, b \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{-i \, \pi ^{3} - 6 \, \pi ^{2} a + 12 i \, \pi a^{2} + 8 \, a^{3}} - \frac {16 \, b \log \left (x\right )}{-i \, \pi ^{3} - 6 \, \pi ^{2} a + 12 i \, \pi a^{2} + 8 \, a^{3}} + \frac {8 \, b}{2 \, \pi ^{2} b x - 8 i \, \pi a b x - 8 \, a^{2} b x + i \, \pi ^{3} + 6 \, \pi ^{2} a - 12 i \, \pi a^{2} - 8 \, a^{3}} + \frac {4}{\pi ^{2} x - 4 i \, \pi a x - 4 \, a^{2} x} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{x^{2} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]________________________________________________________________________________________