\[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {16 \cot \left (d x +c \right ) \sqrt {e \csc \left (d x +c \right )}}{21 a^{2} d}-\frac {2 \left (\cot ^{3}\left (d x +c \right )\right ) \sqrt {e \csc \left (d x +c \right )}}{7 a^{2} d}-\frac {4 \csc \left (d x +c \right ) \sqrt {e \csc \left (d x +c \right )}}{3 a^{2} d}-\frac {2 \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right ) \sqrt {e \csc \left (d x +c \right )}}{7 a^{2} d}+\frac {4 \left (\csc ^{3}\left (d x +c \right )\right ) \sqrt {e \csc \left (d x +c \right )}}{7 a^{2} d}-\frac {20 \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \sqrt {e \csc \left (d x +c \right )}\, \left (\sqrt {\sin }\left (d x +c \right )\right )}{21 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) a^{2} d} \]
command
integrate((e*csc(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
\[ -\frac {2 \, {\left (5 \, \sqrt {2 i} {\left (i \, \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} + 2 i \, \cos \left (d x + c\right ) e^{\frac {1}{2}} + i \, e^{\frac {1}{2}}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {-2 i} {\left (-i \, \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 2 i \, \cos \left (d x + c\right ) e^{\frac {1}{2}} - i \, e^{\frac {1}{2}}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (11 \, \cos \left (d x + c\right ) e^{\frac {1}{2}} + 8 \, e^{\frac {1}{2}}\right )} \sqrt {\sin \left (d x + c\right )}\right )}}{21 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )}}{a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}}, x\right ) \]