\[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {i b^{2} f \polylog \left (2, i {\mathrm e}^{d x +c}\right )}{2 a \left (a^{2}+b^{2}\right ) d^{2}}+\frac {i b^{4} f \polylog \left (2, i {\mathrm e}^{d x +c}\right )}{a \left (a^{2}+b^{2}\right )^{2} d^{2}}-\frac {i b^{4} f \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \left (a^{2}+b^{2}\right )^{2} d^{2}}-\frac {i b^{2} f \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{2 a \left (a^{2}+b^{2}\right ) d^{2}}+\frac {b^{2} f \,\mathrm {sech}\left (d x +c \right )}{2 a \left (a^{2}+b^{2}\right ) d^{2}}+\frac {b^{3} \left (f x +e \right ) \mathrm {sech}\left (d x +c \right )^{2}}{2 a^{2} \left (a^{2}+b^{2}\right ) d}-\frac {b^{3} f \tanh \left (d x +c \right )}{2 a^{2} \left (a^{2}+b^{2}\right ) d^{2}}+\frac {b^{2} \left (f x +e \right ) \arctan \left ({\mathrm e}^{d x +c}\right )}{a \left (a^{2}+b^{2}\right ) d}+\frac {2 b^{4} \left (f x +e \right ) \arctan \left ({\mathrm e}^{d x +c}\right )}{a \left (a^{2}+b^{2}\right )^{2} d}-\frac {f \,\mathrm {sech}\left (d x +c \right )}{2 a \,d^{2}}-\frac {b \left (f x +e \right ) \ln \left (\tanh \left (d x +c \right )\right )}{a^{2} d}-\frac {3 \left (f x +e \right ) \mathrm {csch}\left (d x +c \right )}{2 a d}-\frac {3 \left (f x +e \right ) \arctan \left (\sinh \left (d x +c \right )\right )}{2 a d}+\frac {\left (f x +e \right ) \mathrm {csch}\left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{2}}{2 a d}+\frac {b f \tanh \left (d x +c \right )}{2 a^{2} d^{2}}-\frac {3 f x \arctan \left ({\mathrm e}^{d x +c}\right )}{a d}+\frac {3 f x \arctan \left (\sinh \left (d x +c \right )\right )}{2 a d}-\frac {f \arctanh \left (\cosh \left (d x +c \right )\right )}{a \,d^{2}}-\frac {b^{5} f \polylog \left (2, -{\mathrm e}^{2 d x +2 c}\right )}{2 a^{2} \left (a^{2}+b^{2}\right )^{2} d^{2}}+\frac {2 b f x \arctanh \left ({\mathrm e}^{2 d x +2 c}\right )}{a^{2} d}+\frac {b^{5} f \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a -\sqrt {a^{2}+b^{2}}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2} d^{2}}+\frac {b^{5} f \polylog \left (2, -\frac {b \,{\mathrm e}^{d x +c}}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2} d^{2}}-\frac {b f \polylog \left (2, {\mathrm e}^{2 d x +2 c}\right )}{2 a^{2} d^{2}}+\frac {b f \polylog \left (2, -{\mathrm e}^{2 d x +2 c}\right )}{2 a^{2} d^{2}}+\frac {3 i f \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{2 a \,d^{2}}-\frac {3 i f \polylog \left (2, i {\mathrm e}^{d x +c}\right )}{2 a \,d^{2}}+\frac {b \left (f x +e \right ) \left (\tanh ^{2}\left (d x +c \right )\right )}{2 a^{2} d}-\frac {b f x}{2 a^{2} d}-\frac {b^{5} \left (f x +e \right ) \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2} d}+\frac {b^{5} \left (f x +e \right ) \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a -\sqrt {a^{2}+b^{2}}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2} d}+\frac {b^{5} \left (f x +e \right ) \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2} d}+\frac {b f x \ln \left (\tanh \left (d x +c \right )\right )}{a^{2} d}+\frac {b^{2} \left (f x +e \right ) \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2 a \left (a^{2}+b^{2}\right ) d} \]
command
integrate((f*x+e)*csch(d*x+c)^2*sech(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 \[ \text {Timed out} \]