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