\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {26 \cot \left (x \right )}{77 a^{2} \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}-\frac {26 \sqrt {\frac {1}{2}+\frac {\sin \left (x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {x}{2}\right ), \sqrt {2}\right )}{77 \sin \left (\frac {\pi }{4}+\frac {x}{2}\right ) a^{2} \sin \left (x \right )^{\frac {3}{2}} \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}-\frac {78 \cos \left (x \right ) \sin \left (x \right )}{385 a^{2} \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}-\frac {26 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )}{165 a^{2} \sqrt {a \left (\csc ^{3}\left (x \right )\right )}}-\frac {2 \cos \left (x \right ) \left (\sin ^{5}\left (x \right )\right )}{15 a^{2} \sqrt {a \left (\csc ^{3}\left (x \right )\right )}} \]
command
integrate(1/(a*csc(x)^3)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (77 \, \cos \left (x\right )^{9} - 399 \, \cos \left (x\right )^{7} + 852 \, \cos \left (x\right )^{5} - 1010 \, \cos \left (x\right )^{3} + 480 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} + 195 i \, \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 195 i \, \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )}{1155 \, a^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \csc \left (x\right )^{3}}}{a^{3} \csc \left (x\right )^{9}}, x\right ) \]