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