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