\[ \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx \]
Optimal antiderivative \[ -\frac {3 b^{2}}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{3} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}+\frac {2 b}{x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}+\frac {1}{2 x^{2} \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right ) \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}+\frac {6 b^{2}}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{4} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}-\frac {6 b^{2} \ln \left (x \right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{5}}+\frac {6 b^{2} \ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{\left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{5}} \]
command
integrate(1/x^3/arccoth(tanh(b*x+a))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {192 i \, b^{2} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{5} - 10 i \, \pi ^{4} a - 40 \, \pi ^{3} a^{2} + 80 i \, \pi ^{2} a^{3} + 80 \, \pi a^{4} - 32 i \, a^{5}} - \frac {192 i \, b^{2} \log \left (x\right )}{\pi ^{5} - 10 i \, \pi ^{4} a - 40 \, \pi ^{3} a^{2} + 80 i \, \pi ^{2} a^{3} + 80 \, \pi a^{4} - 32 i \, a^{5}} - \frac {4 \, {\left (i \, \pi - 12 \, b x + 2 \, a\right )}}{\pi ^{4} x^{2} - 8 i \, \pi ^{3} a x^{2} - 24 \, \pi ^{2} a^{2} x^{2} + 32 i \, \pi a^{3} x^{2} + 16 \, a^{4} x^{2}} + \frac {16 \, {\left (12 \, b^{3} x + 7 i \, \pi b^{2} + 14 \, a b^{2}\right )}}{4 \, \pi ^{4} b^{2} x^{2} - 32 i \, \pi ^{3} a b^{2} x^{2} - 96 \, \pi ^{2} a^{2} b^{2} x^{2} + 128 i \, \pi a^{3} b^{2} x^{2} + 64 \, a^{4} b^{2} x^{2} + 4 i \, \pi ^{5} b x + 40 \, \pi ^{4} a b x - 160 i \, \pi ^{3} a^{2} b x - 320 \, \pi ^{2} a^{3} b x + 320 i \, \pi a^{4} b x + 128 \, a^{5} b x - \pi ^{6} + 12 i \, \pi ^{5} a + 60 \, \pi ^{4} a^{2} - 160 i \, \pi ^{3} a^{3} - 240 \, \pi ^{2} a^{4} + 192 i \, \pi a^{5} + 64 \, a^{6}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}\,{d x} \]________________________________________________________________________________________