\[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4 e^{3}}{9 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}-\frac {2 e^{3} \cos \left (d x +c \right )}{9 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}-\frac {2 e^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{9 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}-\frac {4 e}{5 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {16 e \cos \left (d x +c \right )}{45 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {4 \cos \left (d x +c \right )}{15 a^{2} d e \sqrt {e \sin \left (d x +c \right )}}+\frac {4 \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \sqrt {e \sin \left (d x +c \right )}}{15 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) a^{2} d \,e^{2} \sqrt {\sin \left (d x +c \right )}} \]
command
integrate(1/(a+a*sec(d*x+c))^2/(e*sin(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (3 \, \sqrt {-i} {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {i} {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (6 \, \cos \left (d x + c\right )^{3} + 12 \, \cos \left (d x + c\right )^{2} + 19 \, \cos \left (d x + c\right ) + 8\right )} \sqrt {\sin \left (d x + c\right )}\right )}}{45 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + 2 \, a^{2} d \cos \left (d x + c\right ) e^{\frac {3}{2}} + a^{2} d e^{\frac {3}{2}}\right )} \sin \left (d x + c\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {e \sin \left (d x + c\right )}}{a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2} + {\left (a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sec \left (d x + c\right )^{2} + 2 \, {\left (a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sec \left (d x + c\right )}, x\right ) \]