\[ \int \sqrt {b \sec (e+f x)} \sin ^6(e+f x) \, dx \]
Optimal antiderivative \[ -\frac {40 b \sin \left (f x +e \right )}{77 f \sqrt {b \sec \left (f x +e \right )}}-\frac {20 b \left (\sin ^{3}\left (f x +e \right )\right )}{77 f \sqrt {b \sec \left (f x +e \right )}}-\frac {2 b \left (\sin ^{5}\left (f x +e \right )\right )}{11 f \sqrt {b \sec \left (f x +e \right )}}+\frac {80 \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{77 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f} \]
command
integrate(sin(f*x+e)^6*(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (7 \, \cos \left (f x + e\right )^{5} - 24 \, \cos \left (f x + e\right )^{3} + 37 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 20 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 20 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{77 \, f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (\cos \left (f x + e\right )^{6} - 3 \, \cos \left (f x + e\right )^{4} + 3 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {b \sec \left (f x + e\right )}, x\right ) \]