\[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {12 b \sin \left (f x +e \right )}{77 f \left (b \sec \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {2 b \left (\sin ^{3}\left (f x +e \right )\right )}{11 f \left (b \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {8 \sin \left (f x +e \right )}{77 b f \sqrt {b \sec \left (f x +e \right )}}+\frac {8 \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{77 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{2} f} \]
command
integrate(sin(f*x+e)^4/(b*sec(f*x+e))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (7 \, \cos \left (f x + e\right )^{5} - 13 \, \cos \left (f x + e\right )^{3} + 4 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 2 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 2 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{77 \, b^{2} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b \sec \left (f x + e\right )}}{b^{2} \sec \left (f x + e\right )^{2}}, x\right ) \]