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