\[ \int \frac {\sinh ^5(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\cosh \left (d x +c \right ) \left (a +b -b \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{8 \left (a -b \right ) b d \left (a -b +2 b \left (\cosh ^{2}\left (d x +c \right )\right )-b \left (\cosh ^{4}\left (d x +c \right )\right )\right )^{2}}-\frac {\cosh \left (d x +c \right ) \left (a^{2}-11 a b -2 b^{2}+2 b \left (2 a +b \right ) \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{32 a \left (a -b \right )^{2} b d \left (a -b +2 b \left (\cosh ^{2}\left (d x +c \right )\right )-b \left (\cosh ^{4}\left (d x +c \right )\right )\right )}-\frac {\arctan \left (\frac {b^{\frac {1}{4}} \cosh \left (d x +c \right )}{\sqrt {\sqrt {a}-\sqrt {b}}}\right ) \left (3 a +4 b -10 \sqrt {a}\, \sqrt {b}\right )}{64 a^{\frac {3}{2}} b^{\frac {5}{4}} d \left (\sqrt {a}-\sqrt {b}\right )^{\frac {5}{2}}}-\frac {\arctanh \left (\frac {b^{\frac {1}{4}} \cosh \left (d x +c \right )}{\sqrt {\sqrt {a}+\sqrt {b}}}\right ) \left (3 a +4 b +10 \sqrt {a}\, \sqrt {b}\right )}{64 a^{\frac {3}{2}} b^{\frac {5}{4}} d \left (\sqrt {a}+\sqrt {b}\right )^{\frac {5}{2}}} \]
command
integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4)^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 \[ \text {Timed out} \]