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