\[ \int \frac {\sinh ^6(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}}\, \tanh \left (d x +c \right )}{a^{\frac {1}{4}}}\right ) \left (4 a +3 b -10 \sqrt {a}\, \sqrt {b}\right )}{64 a^{\frac {5}{4}} b^{\frac {3}{2}} d \left (\sqrt {a}-\sqrt {b}\right )^{\frac {5}{2}}}-\frac {\arctanh \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}}\, \tanh \left (d x +c \right )}{a^{\frac {1}{4}}}\right ) \left (4 a +3 b +10 \sqrt {a}\, \sqrt {b}\right )}{64 a^{\frac {5}{4}} b^{\frac {3}{2}} d \left (\sqrt {a}+\sqrt {b}\right )^{\frac {5}{2}}}+\frac {\tanh \left (d x +c \right ) \left (a \left (a +3 b \right )-\left (a^{2}+6 a b +b^{2}\right ) \left (\tanh ^{2}\left (d x +c \right )\right )\right )}{8 \left (a -b \right )^{3} d \left (a -2 a \left (\tanh ^{2}\left (d x +c \right )\right )+\left (a -b \right ) \left (\tanh ^{4}\left (d x +c \right )\right )\right )^{2}}+\frac {\tanh \left (d x +c \right ) \left (\frac {2 a \left (a^{2}-a b -8 b^{2}\right )}{\left (a -b \right )^{3}}-\frac {\left (2 a^{2}+15 a b +3 b^{2}\right ) \left (\tanh ^{2}\left (d x +c \right )\right )}{\left (a -b \right )^{2}}\right )}{32 a b d \left (a -2 a \left (\tanh ^{2}\left (d x +c \right )\right )+\left (a -b \right ) \left (\tanh ^{4}\left (d x +c \right )\right )\right )} \]
command
integrate(sinh(d*x+c)^6/(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 \[ \text {Timed out} \]