\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {10 \cos \left (x \right )}{21 a \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}-\frac {2 \cot \left (x \right ) \csc \left (x \right )}{7 a \sqrt {a \left (\sin ^{3}\left (x \right )\right )}}-\frac {10 \sqrt {\frac {1}{2}+\frac {\sin \left (x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {x}{2}\right ), \sqrt {2}\right ) \left (\sin ^{\frac {3}{2}}\left (x \right )\right )}{21 \sin \left (\frac {\pi }{4}+\frac {x}{2}\right ) a \sqrt {a \left (\sin ^{3}\left (x \right )\right )}} \]
command
integrate(1/(a*sin(x)^3)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {5 \, {\left (\sqrt {2} \cos \left (x\right )^{4} - 2 \, \sqrt {2} \cos \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {-i \, a} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 5 \, {\left (\sqrt {2} \cos \left (x\right )^{4} - 2 \, \sqrt {2} \cos \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {i \, a} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (5 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{21 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \sin \left (x\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{a^{2} \cos \left (x\right )^{6} - 3 \, a^{2} \cos \left (x\right )^{4} + 3 \, a^{2} \cos \left (x\right )^{2} - a^{2}}, x\right ) \]