\[ \int (b \sec (e+f x))^{3/2} \sin ^4(e+f x) \, dx \]
Optimal antiderivative \[ \frac {12 b^{3} \sin \left (f x +e \right )}{5 f \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {24 b^{2} \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f \sqrt {\cos \left (f x +e \right )}\, \sqrt {b \sec \left (f x +e \right )}}+\frac {2 b \left (\sin ^{3}\left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{f} \]
command
integrate((b*sec(f*x+e))^(3/2)*sin(f*x+e)^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (6 i \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 6 i \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (b \cos \left (f x + e\right )^{2} + 5 \, b\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )\right )}}{5 \, f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b \cos \left (f x + e\right )^{4} - 2 \, b \cos \left (f x + e\right )^{2} + b\right )} \sqrt {b \sec \left (f x + e\right )} \sec \left (f x + e\right ), x\right ) \]