\[ \int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {b^{2} \left (f x +e \right )^{4}}{4 a^{3} f}+\frac {\left (a^{2}+b^{2}\right ) \left (f x +e \right )^{4}}{4 a^{3} f}-\frac {\left (a^{2}+b^{2}\right ) \left (f x +e \right )^{3} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a -\sqrt {a^{2}+b^{2}}}\right )}{a^{3} d}-\frac {\left (a^{2}+b^{2}\right ) \left (f x +e \right )^{3} \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{3} d}+\frac {b \left (f x +e \right )^{3} \mathrm {csch}\left (d x +c \right )}{a^{2} d}-\frac {3 f \left (f x +e \right )^{2} \coth \left (d x +c \right )}{2 a \,d^{2}}-\frac {3 \left (a^{2}+b^{2}\right ) f \left (f x +e \right )^{2} \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a -\sqrt {a^{2}+b^{2}}}\right )}{a^{3} d^{2}}+\frac {6 b f \left (f x +e \right )^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )}{a^{2} d^{2}}+\frac {6 b \,f^{2} \left (f x +e \right ) \polylog \left (2, -{\mathrm e}^{d x +c}\right )}{a^{2} d^{3}}-\frac {6 b \,f^{2} \left (f x +e \right ) \polylog \left (2, {\mathrm e}^{d x +c}\right )}{a^{2} d^{3}}+\frac {3 b^{2} f \left (f x +e \right )^{2} \polylog \left (2, {\mathrm e}^{2 d x +2 c}\right )}{2 a^{3} d^{2}}-\frac {3 \left (a^{2}+b^{2}\right ) f \left (f x +e \right )^{2} \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{3} d^{2}}+\frac {6 \left (a^{2}+b^{2}\right ) f^{2} \left (f x +e \right ) \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a -\sqrt {a^{2}+b^{2}}}\right )}{a^{3} d^{3}}+\frac {6 \left (a^{2}+b^{2}\right ) f^{2} \left (f x +e \right ) \polylog \left (3, -\frac {b \,{\mathrm e}^{d x +c}}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{3} d^{3}}-\frac {3 b^{2} f^{2} \left (f x +e \right ) \polylog \left (3, {\mathrm e}^{2 d x +2 c}\right )}{2 a^{3} d^{3}}-\frac {3 f \left (f x +e \right )^{2}}{2 a \,d^{2}}-\frac {\left (f x +e \right )^{4}}{4 a f}+\frac {\left (f x +e \right )^{3}}{2 a d}+\frac {\left (f x +e \right )^{3} \ln \left (1-{\mathrm e}^{2 d x +2 c}\right )}{a d}+\frac {3 f^{2} \left (f x +e \right ) \ln \left (1-{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{3}}+\frac {b^{2} \left (f x +e \right )^{3} \ln \left (1-{\mathrm e}^{2 d x +2 c}\right )}{a^{3} d}+\frac {3 f^{3} \polylog \left (2, {\mathrm e}^{2 d x +2 c}\right )}{2 a \,d^{4}}+\frac {3 f^{3} \polylog \left (4, {\mathrm e}^{2 d x +2 c}\right )}{4 a \,d^{4}}-\frac {\left (f x +e \right )^{3} \left (\coth ^{2}\left (d x +c \right )\right )}{2 a d}-\frac {6 \left (a^{2}+b^{2}\right ) f^{3} \polylog \left (4, -\frac {b \,{\mathrm e}^{d x +c}}{a -\sqrt {a^{2}+b^{2}}}\right )}{a^{3} d^{4}}-\frac {6 \left (a^{2}+b^{2}\right ) f^{3} \polylog \left (4, -\frac {b \,{\mathrm e}^{d x +c}}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{3} d^{4}}-\frac {6 b \,f^{3} \polylog \left (3, -{\mathrm e}^{d x +c}\right )}{a^{2} d^{4}}+\frac {6 b \,f^{3} \polylog \left (3, {\mathrm e}^{d x +c}\right )}{a^{2} d^{4}}+\frac {3 b^{2} f^{3} \polylog \left (4, {\mathrm e}^{2 d x +2 c}\right )}{4 a^{3} d^{4}}+\frac {3 f \left (f x +e \right )^{2} \polylog \left (2, {\mathrm e}^{2 d x +2 c}\right )}{2 a \,d^{2}}-\frac {3 f^{2} \left (f x +e \right ) \polylog \left (3, {\mathrm e}^{2 d x +2 c}\right )}{2 a \,d^{3}} \]
command
integrate((f*x+e)^3*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]