\[ \int \frac {1}{\sqrt {\sin (x)} \sqrt {a-a \sin (x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\cos \left (x \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {\sin \left (x \right )}\, \sqrt {a -a \sin \left (x \right )}}\right ) \sqrt {2}}{\sqrt {a}} \]
command
integrate(1/sin(x)^(1/2)/(a-a*sin(x))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (\log \left (\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} - \sqrt {\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{4} - 6 \, \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + 1} + 1\right ) - \log \left ({\left | -\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + \sqrt {\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{4} - 6 \, \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + 1} + 3 \right |}\right ) - \log \left ({\left | -\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + \sqrt {\tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{4} - 6 \, \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, x\right )^{2} + 1} + 1 \right |}\right )\right )}}{2 \, \sqrt {a} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {-a \sin \left (x\right ) + a} \sqrt {\sin \left (x\right )}}\,{d x} \]________________________________________________________________________________________