\[ \int \frac {\csc ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {7 b \csc \left (f x +e \right )}{20 f \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {7 b \left (\csc ^{3}\left (f x +e \right )\right )}{30 f \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {b \left (\csc ^{5}\left (f x +e \right )\right )}{5 f \left (b \sec \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {7 \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right )}{20 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f \sqrt {\cos \left (f x +e \right )}\, \sqrt {b \sec \left (f x +e \right )}} \]
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
\[ -\frac {21 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{4} - 2 i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{4} + 2 i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (21 \, \cos \left (f x + e\right )^{6} - 56 \, \cos \left (f x + e\right )^{4} + 47 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{120 \, {\left (b f \cos \left (f x + e\right )^{4} - 2 \, b f \cos \left (f x + e\right )^{2} + b f\right )} \sin \left (f x + e\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{6}}{b \sec \left (f x + e\right )}, x\right ) \]