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