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