\[ \int \frac {\sinh ^9(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ \frac {a \cosh \left (d x +c \right ) \left (a +b -b \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{8 \left (a -b \right ) b^{2} d \left (a -b +2 b \left (\cosh ^{2}\left (d x +c \right )\right )-b \left (\cosh ^{4}\left (d x +c \right )\right )\right )^{2}}-\frac {\cosh \left (d x +c \right ) \left (9 a^{2}-11 a b -10 b^{2}-2 \left (2 a -5 b \right ) b \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{32 \left (a -b \right )^{2} b^{2} d \left (a -b +2 b \left (\cosh ^{2}\left (d x +c \right )\right )-b \left (\cosh ^{4}\left (d x +c \right )\right )\right )}+\frac {\arctan \left (\frac {b^{\frac {1}{4}} \cosh \left (d x +c \right )}{\sqrt {\sqrt {a}-\sqrt {b}}}\right ) \left (5 a +12 b -14 \sqrt {a}\, \sqrt {b}\right )}{64 b^{\frac {9}{4}} d \sqrt {a}\, \left (\sqrt {a}-\sqrt {b}\right )^{\frac {5}{2}}}+\frac {\arctanh \left (\frac {b^{\frac {1}{4}} \cosh \left (d x +c \right )}{\sqrt {\sqrt {a}+\sqrt {b}}}\right ) \left (5 a +12 b +14 \sqrt {a}\, \sqrt {b}\right )}{64 b^{\frac {9}{4}} d \sqrt {a}\, \left (\sqrt {a}+\sqrt {b}\right )^{\frac {5}{2}}} \]
command
integrate(sinh(d*x+c)^9/(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 \[ \text {Timed out} \]