\[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \sin (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {4 e^{3}}{7 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {2 e^{3} \cos \left (d x +c \right )}{7 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {2 e^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{7 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {4 e}{3 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {16 e \cos \left (d x +c \right )}{21 a^{2} d \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {20 \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\sin }\left (d x +c \right )\right )}{21 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) a^{2} d \sqrt {e \sin \left (d x +c \right )}} \]
command
integrate(1/(a+a*sec(d*x+c))^2/(e*sin(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (5 \, \sqrt {-i} {\left (\sqrt {2} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {i} {\left (\sqrt {2} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - {\left (11 \, \cos \left (d x + c\right ) + 8\right )} \sqrt {\sin \left (d x + c\right )}\right )}}{21 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} + 2 \, a^{2} d \cos \left (d x + c\right ) e^{\frac {1}{2}} + a^{2} d e^{\frac {1}{2}}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \sin \left (d x + c\right )}}{{\left (a^{2} e \sec \left (d x + c\right )^{2} + 2 \, a^{2} e \sec \left (d x + c\right ) + a^{2} e\right )} \sin \left (d x + c\right )}, x\right ) \]