\[ \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {3 b \csc \left (f x +e \right )}{4 f \sqrt {b \sec \left (f x +e \right )}}-\frac {3 b \left (\csc ^{3}\left (f x +e \right )\right )}{10 f \sqrt {b \sec \left (f x +e \right )}}-\frac {b \left (\csc ^{5}\left (f x +e \right )\right )}{5 f \sqrt {b \sec \left (f x +e \right )}}+\frac {3 \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f} \]
command
integrate(csc(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
\[ -\frac {15 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{4} - 2 i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 15 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{4} + 2 i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (15 \, \cos \left (f x + e\right )^{5} - 36 \, \cos \left (f x + e\right )^{3} + 25 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{40 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )} \sin \left (f x + e\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{6}, x\right ) \]