\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {154 \cot \left (x \right )}{585 a^{2} \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}-\frac {22 \cot \left (x \right ) \left (\csc ^{2}\left (x \right )\right )}{117 a^{2} \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}-\frac {2 \cot \left (x \right ) \left (\csc ^{4}\left (x \right )\right )}{13 a^{2} \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}-\frac {154 \cos \left (x \right ) \sin \left (x \right )}{195 a^{2} \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}+\frac {154 \sqrt {\frac {1}{2}+\frac {\sin \left (x \right )}{2}}\, \EllipticE \left (\cos \left (\frac {\pi }{4}+\frac {x}{2}\right ), \sqrt {2}\right ) \left (\sin ^{\frac {3}{2}}\left (x \right )\right )}{195 \sin \left (\frac {\pi }{4}+\frac {x}{2}\right ) a^{2} \sqrt {a \left (\sin ^{3}\left (x \right )\right )}} \]
command
integrate(1/(a*sin(x)^3)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {231 \, {\left (i \, \sqrt {2} \cos \left (x\right )^{8} - 4 i \, \sqrt {2} \cos \left (x\right )^{6} + 6 i \, \sqrt {2} \cos \left (x\right )^{4} - 4 i \, \sqrt {2} \cos \left (x\right )^{2} + i \, \sqrt {2}\right )} \sqrt {-i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + 231 \, {\left (-i \, \sqrt {2} \cos \left (x\right )^{8} + 4 i \, \sqrt {2} \cos \left (x\right )^{6} - 6 i \, \sqrt {2} \cos \left (x\right )^{4} + 4 i \, \sqrt {2} \cos \left (x\right )^{2} - i \, \sqrt {2}\right )} \sqrt {i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) - 2 \, {\left (231 \, \cos \left (x\right )^{7} - 770 \, \cos \left (x\right )^{5} + 902 \, \cos \left (x\right )^{3} - 408 \, \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{585 \, {\left (a^{3} \cos \left (x\right )^{8} - 4 \, a^{3} \cos \left (x\right )^{6} + 6 \, a^{3} \cos \left (x\right )^{4} - 4 \, a^{3} \cos \left (x\right )^{2} + a^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{{\left (a^{3} \cos \left (x\right )^{8} - 4 \, a^{3} \cos \left (x\right )^{6} + 6 \, a^{3} \cos \left (x\right )^{4} - 4 \, a^{3} \cos \left (x\right )^{2} + a^{3}\right )} \sin \left (x\right )}, x\right ) \]