\[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (a -7 b \right ) \arctan \left (\sinh \left (d x +c \right )\right )}{2 \left (a -b \right )^{4} d}+\frac {b^{\frac {3}{2}} \left (35 a^{2}-14 a b +3 b^{2}\right ) \arctan \left (\frac {\sinh \left (d x +c \right ) \sqrt {b}}{\sqrt {a}}\right )}{8 a^{\frac {5}{2}} \left (a -b \right )^{4} d}+\frac {b \left (2 a +b \right ) \sinh \left (d x +c \right )}{4 a \left (a -b \right )^{2} d \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )^{2}}+\frac {\left (4 a -b \right ) b \left (a +3 b \right ) \sinh \left (d x +c \right )}{8 a^{2} \left (a -b \right )^{3} d \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )}+\frac {\mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2 \left (a -b \right ) d \left (a +b \left (\sinh ^{2}\left (d x +c \right )\right )\right )^{2}} \]
command
integrate(sech(d*x+c)^3/(a+b*sinh(d*x+c)^2)^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 \[ \text {Timed out} \]