\[ \int \frac {\sinh ^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}}\, \tanh \left (d x +c \right )}{a^{\frac {1}{4}}}\right ) \left (12 a +5 b -14 \sqrt {a}\, \sqrt {b}\right )}{64 a^{\frac {9}{4}} d \left (\sqrt {a}-\sqrt {b}\right )^{\frac {5}{2}} \sqrt {b}}+\frac {\arctanh \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}}\, \tanh \left (d x +c \right )}{a^{\frac {1}{4}}}\right ) \left (12 a +5 b +14 \sqrt {a}\, \sqrt {b}\right )}{64 a^{\frac {9}{4}} d \sqrt {b}\, \left (\sqrt {a}+\sqrt {b}\right )^{\frac {5}{2}}}+\frac {b \tanh \left (d x +c \right ) \left (a \left (a +3 b \right )-\left (a^{2}+6 a b +b^{2}\right ) \left (\tanh ^{2}\left (d x +c \right )\right )\right )}{8 a \left (a -b \right )^{3} d \left (a -2 a \left (\tanh ^{2}\left (d x +c \right )\right )+\left (a -b \right ) \left (\tanh ^{4}\left (d x +c \right )\right )\right )^{2}}+\frac {\tanh \left (d x +c \right ) \left (\frac {2 a \left (5 a^{2}-9 a b -4 b^{2}\right )}{\left (a -b \right )^{3}}-\frac {5 \left (2 a^{2}+3 a b -b^{2}\right ) \left (\tanh ^{2}\left (d x +c \right )\right )}{\left (a -b \right )^{2}}\right )}{32 a^{2} d \left (a -2 a \left (\tanh ^{2}\left (d x +c \right )\right )+\left (a -b \right ) \left (\tanh ^{4}\left (d x +c \right )\right )\right )} \]
command
integrate(sinh(d*x+c)^2/(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 \[ \text {Timed out} \]