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