\[ \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {7 b^{2} \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right )}{2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) f \sqrt {\cos \left (f x +e \right )}\, \sqrt {b \sec \left (f x +e \right )}}-\frac {7 b \csc \left (f x +e \right ) \sqrt {b \sec \left (f x +e \right )}}{6 f}-\frac {b \left (\csc ^{3}\left (f x +e \right )\right ) \sqrt {b \sec \left (f x +e \right )}}{3 f}+\frac {7 b \sin \left (f x +e \right ) \sqrt {b \sec \left (f x +e \right )}}{2 f} \]
command
integrate(csc(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
\[ -\frac {21 \, \sqrt {2} {\left (i \, b \cos \left (f x + e\right )^{2} - i \, b\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, b \cos \left (f x + e\right )^{2} + i \, b\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (21 \, b \cos \left (f x + e\right )^{4} - 35 \, b \cos \left (f x + e\right )^{2} + 12 \, b\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b \sec \left (f x + e\right )} b \csc \left (f x + e\right )^{4} \sec \left (f x + e\right ), x\right ) \]