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