\[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {4 e^{3}}{5 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {2 e^{3} \cos \left (d x +c \right )}{5 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {2 e^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{5 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {4 e}{a^{2} d \sqrt {e \sin \left (d x +c \right )}}+\frac {16 e \cos \left (d x +c \right )}{5 a^{2} d \sqrt {e \sin \left (d x +c \right )}}-\frac {28 \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \sqrt {e \sin \left (d x +c \right )}}{5 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) a^{2} d \sqrt {\sin \left (d x +c \right )}} \]
command
integrate((e*sin(d*x+c))^(1/2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (9 \, \cos \left (d x + c\right ) e^{\frac {1}{2}} + 8 \, e^{\frac {1}{2}}\right )} \sin \left (d x + c\right )^{\frac {3}{2}} + 7 \, \sqrt {-i} {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 2 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} - i \, \sqrt {2} e^{\frac {1}{2}}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 7 \, \sqrt {i} {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} + 2 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} + i \, \sqrt {2} e^{\frac {1}{2}}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )\right )}}{5 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \sin \left (d x + c\right )}}{a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}}, x\right ) \]