\[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (a +b \right ) \sqrt {2}\, \sqrt {\frac {1}{1+\cosh \left (2 f x +2 e \right )}}\, \sqrt {1+\sinh ^{2}\left (f x +e \right )}\, \EllipticE \left (\frac {\sinh \left (f x +e \right )}{\sqrt {1+\sinh ^{2}\left (f x +e \right )}}, \sqrt {1-\frac {b}{a}}\right ) \mathrm {sech}\left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{3 f \sqrt {\frac {\mathrm {sech}\left (f x +e \right )^{2} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )}{a}}}-\frac {b \sqrt {2}\, \sqrt {\frac {1}{1+\cosh \left (2 f x +2 e \right )}}\, \sqrt {1+\sinh ^{2}\left (f x +e \right )}\, \EllipticF \left (\frac {\sinh \left (f x +e \right )}{\sqrt {1+\sinh ^{2}\left (f x +e \right )}}, \sqrt {1-\frac {b}{a}}\right ) \mathrm {sech}\left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{3 f \sqrt {\frac {\mathrm {sech}\left (f x +e \right )^{2} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )}{a}}}+\frac {\left (a -b \right ) \mathrm {sech}\left (f x +e \right )^{2} \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, \tanh \left (f x +e \right )}{3 f} \]
command
integrate(sech(f*x+e)^4*(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 \[ \text {Exception raised: TypeError} \]_______________________________________________________