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