\[ \int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx \]
Optimal antiderivative \[ -\frac {2 d^{4} \cos \left (f x +e \right )}{7 f \left (d \csc \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {10 d^{2} \cos \left (f x +e \right )}{21 f \sqrt {d \csc \left (f x +e \right )}}-\frac {10 d \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\right ) \sqrt {d \csc \left (f x +e \right )}\, \left (\sqrt {\sin }\left (f x +e \right )\right )}{21 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) f} \]
command
integrate((d*csc(f*x+e))^(3/2)*sin(f*x+e)^5,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (3 \, d \cos \left (f x + e\right )^{3} - 8 \, d \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}} \sin \left (f x + e\right ) - 5 i \, \sqrt {2 i \, d} d {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {-2 i \, d} d {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{21 \, f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (d \cos \left (f x + e\right )^{4} - 2 \, d \cos \left (f x + e\right )^{2} + d\right )} \sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right ) \sin \left (f x + e\right ), x\right ) \]