\[ \int \frac {\sinh ^5(c+d x)}{a+b \sinh ^3(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {x}{2 b}+\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2 b d}+\frac {2 a \arctan \left (\frac {\left (-1\right )^{\frac {1}{6}} \left (\left (-1\right )^{\frac {5}{6}} b^{\frac {1}{3}}+i a^{\frac {1}{3}} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\sqrt {\left (-1\right )^{\frac {1}{3}} a^{\frac {2}{3}}-b^{\frac {2}{3}}}}\right )}{3 b^{\frac {5}{3}} d \sqrt {\left (-1\right )^{\frac {1}{3}} a^{\frac {2}{3}}-b^{\frac {2}{3}}}}+\frac {2 a \arctan \left (\frac {\left (-1\right )^{\frac {5}{6}} \left (\left (-1\right )^{\frac {1}{6}} b^{\frac {1}{3}}+i a^{\frac {1}{3}} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\sqrt {-\left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}}-b^{\frac {2}{3}}}}\right )}{3 b^{\frac {5}{3}} d \sqrt {-\left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}}-b^{\frac {2}{3}}}}+\frac {2 a \arctanh \left (\frac {b^{\frac {1}{3}}-a^{\frac {1}{3}} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {a^{\frac {2}{3}}+b^{\frac {2}{3}}}}\right )}{3 b^{\frac {5}{3}} d \sqrt {a^{\frac {2}{3}}+b^{\frac {2}{3}}}} \]
command
integrate(sinh(d*x+c)^5/(a+b*sinh(d*x+c)^3),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 \[ \text {Timed out} \]