\[ \int \text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {3 a^{2} \arctan \left (\frac {\sinh \left (f x +e \right ) \sqrt {a -b}}{\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}\right )}{8 f \sqrt {a -b}}+\frac {\mathrm {sech}\left (f x +e \right )^{3} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \tanh \left (f x +e \right )}{4 f}+\frac {3 a \,\mathrm {sech}\left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, \tanh \left (f x +e \right )}{8 f} \]
command
integrate(sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: TypeError} \]_______________________________________________________