\[ \int \frac {\csc ^5(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \cos \left (f x +e \right ) \left (d \csc \left (f x +e \right )\right )^{\frac {5}{2}}}{5 d^{4} f}-\frac {6 \cos \left (f x +e \right ) \sqrt {d \csc \left (f x +e \right )}}{5 d^{2} f}+\frac {6 \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\right )}{5 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d f \sqrt {d \csc \left (f x +e \right )}\, \sqrt {\sin \left (f x +e \right )}} \]
command
integrate(csc(f*x+e)^5/(d*csc(f*x+e))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {-2 i \, d} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}}}{5 \, {\left (d^{2} f \cos \left (f x + e\right )^{2} - d^{2} f\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right )^{3}}{d^{2}}, x\right ) \]