\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {14 \cos \left (x \right )}{45 a \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}-\frac {14 \sqrt {\frac {1}{2}+\frac {\sin \left (x \right )}{2}}\, \EllipticE \left (\cos \left (\frac {\pi }{4}+\frac {x}{2}\right ), \sqrt {2}\right )}{15 \sin \left (\frac {\pi }{4}+\frac {x}{2}\right ) a \sin \left (x \right )^{\frac {3}{2}} \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}-\frac {2 \cos \left (x \right ) \left (\sin ^{2}\left (x \right )\right )}{9 a \sqrt {a \left (\csc ^{3}\left (x \right )\right )}} \]
command
integrate(1/(a*csc(x)^3)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (5 \, \cos \left (x\right )^{5} - 17 \, \cos \left (x\right )^{3} + 12 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} \sin \left (x\right ) - 21 \, \sqrt {2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - 21 \, \sqrt {-2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right )}{45 \, a^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \csc \left (x\right )^{3}}}{a^{2} \csc \left (x\right )^{6}}, x\right ) \]