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