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