\[ \int \text {sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (5 a -6 b \right ) \arctan \left (\frac {\sinh \left (f x +e \right ) \sqrt {a -b}}{\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}\right )}{16 \left (a -b \right )^{\frac {3}{2}} f}+\frac {\left (5 a -6 b \right ) \mathrm {sech}\left (f x +e \right )^{3} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \tanh \left (f x +e \right )}{24 \left (a -b \right ) f}+\frac {\mathrm {sech}\left (f x +e \right )^{5} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}} \tanh \left (f x +e \right )}{6 \left (a -b \right ) f}+\frac {a \left (5 a -6 b \right ) \mathrm {sech}\left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, \tanh \left (f x +e \right )}{16 \left (a -b \right ) f} \]
command
integrate(sech(f*x+e)^7*(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 \[ \text {Exception raised: TypeError} \]_______________________________________________________