\[ \int \frac {\sinh ^4(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ \frac {3 \arctanh \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}}\, \tanh \left (d x +c \right )}{a^{\frac {1}{4}}}\right ) \left (2 \sqrt {a}-\sqrt {b}\right )}{64 a^{\frac {7}{4}} d \left (\sqrt {a}-\sqrt {b}\right )^{\frac {5}{2}} \sqrt {b}}-\frac {3 \arctanh \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}}\, \tanh \left (d x +c \right )}{a^{\frac {1}{4}}}\right ) \left (2 \sqrt {a}+\sqrt {b}\right )}{64 a^{\frac {7}{4}} d \sqrt {b}\, \left (\sqrt {a}+\sqrt {b}\right )^{\frac {5}{2}}}-\frac {b \tanh \left (d x +c \right ) \left (3 a +b -4 \left (a +b \right ) \left (\tanh ^{2}\left (d x +c \right )\right )\right )}{8 \left (a -b \right )^{3} d \left (a -2 a \left (\tanh ^{2}\left (d x +c \right )\right )+\left (a -b \right ) \left (\tanh ^{4}\left (d x +c \right )\right )\right )^{2}}-\frac {\tanh \left (d x +c \right ) \left (\frac {9 a^{2}-24 a b -b^{2}}{\left (a -b \right )^{3}}-\frac {\left (17 a +3 b \right ) \left (\tanh ^{2}\left (d x +c \right )\right )}{\left (a -b \right )^{2}}\right )}{32 a d \left (a -2 a \left (\tanh ^{2}\left (d x +c \right )\right )+\left (a -b \right ) \left (\tanh ^{4}\left (d x +c \right )\right )\right )} \]
command
integrate(sinh(d*x+c)^4/(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 \[ \text {Timed out} \]