\[ \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx \]
Optimal antiderivative \[ -\frac {3 b}{2 \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}+\frac {1}{x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right ) \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}+\frac {3 b}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{3} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}-\frac {3 b \ln \left (x \right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{4}}+\frac {3 b \ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{4}} \]
command
integrate(1/x^2/arccoth(tanh(b*x+a))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {48 i \, b \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{i \, \pi ^{4} + 8 \, \pi ^{3} a - 24 i \, \pi ^{2} a^{2} - 32 \, \pi a^{3} + 16 i \, a^{4}} - \frac {48 i \, b \log \left (x\right )}{i \, \pi ^{4} + 8 \, \pi ^{3} a - 24 i \, \pi ^{2} a^{2} - 32 \, \pi a^{3} + 16 i \, a^{4}} + \frac {16 \, {\left (8 \, b^{2} x + 5 i \, \pi b + 10 \, a b\right )}}{8 i \, \pi ^{3} b^{2} x^{2} + 48 \, \pi ^{2} a b^{2} x^{2} - 96 i \, \pi a^{2} b^{2} x^{2} - 64 \, a^{3} b^{2} x^{2} - 8 \, \pi ^{4} b x + 64 i \, \pi ^{3} a b x + 192 \, \pi ^{2} a^{2} b x - 256 i \, \pi a^{3} b x - 128 \, a^{4} b x - 2 i \, \pi ^{5} - 20 \, \pi ^{4} a + 80 i \, \pi ^{3} a^{2} + 160 \, \pi ^{2} a^{3} - 160 i \, \pi a^{4} - 64 \, a^{5}} + \frac {8}{i \, \pi ^{3} x + 6 \, \pi ^{2} a x - 12 i \, \pi a^{2} x - 8 \, a^{3} x} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{x^{2} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}\,{d x} \]________________________________________________________________________________________