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