\[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {154 a^{2} \cot \left (x \right ) \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}{585}-\frac {22 a^{2} \cot \left (x \right ) \left (\csc ^{2}\left (x \right )\right ) \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}{117}-\frac {2 a^{2} \cot \left (x \right ) \left (\csc ^{4}\left (x \right )\right ) \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}{13}-\frac {154 a^{2} \cos \left (x \right ) \sin \left (x \right ) \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}{195}+\frac {154 a^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (x \right )}{2}}\, \EllipticE \left (\cos \left (\frac {\pi }{4}+\frac {x}{2}\right ), \sqrt {2}\right ) \left (\sin ^{\frac {3}{2}}\left (x \right )\right ) \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}{195 \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )} \]
command
integrate((a*csc(x)^3)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {231 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \sqrt {2 i \, a} \sin \left (x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + 231 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \sqrt {-2 i \, a} \sin \left (x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) - 2 \, {\left (231 \, a^{2} \cos \left (x\right )^{7} - 770 \, a^{2} \cos \left (x\right )^{5} + 902 \, a^{2} \cos \left (x\right )^{3} - 408 \, a^{2} \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}}{585 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \csc \left (x\right )^{3}} a^{2} \csc \left (x\right )^{6}, x\right ) \]