\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx \]
Optimal antiderivative \[ \frac {10 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{77 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d \sqrt {e \cos \left (d x +c \right )}}-\frac {2 \sqrt {e \cos \left (d x +c \right )}}{11 d e \left (a +a \sin \left (d x +c \right )\right )^{3}}-\frac {10 \sqrt {e \cos \left (d x +c \right )}}{77 a d e \left (a +a \sin \left (d x +c \right )\right )^{2}}-\frac {10 \sqrt {e \cos \left (d x +c \right )}}{77 d e \left (a^{3}+a^{3} \sin \left (d x +c \right )\right )} \]
command
integrate(1/(a+a*sin(d*x+c))^3/(e*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, {\left (3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) - 4 i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (-3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) + 4 i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 15 \, \sin \left (d x + c\right ) - 22\right )} \sqrt {\cos \left (d x + c\right )}}{77 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 4 \, a^{3} d e^{\frac {1}{2}} + {\left (a^{3} d \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 4 \, a^{3} d e^{\frac {1}{2}}\right )} \sin \left (d x + c\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {e \cos \left (d x + c\right )}}{3 \, a^{3} e \cos \left (d x + c\right )^{3} - 4 \, a^{3} e \cos \left (d x + c\right ) + {\left (a^{3} e \cos \left (d x + c\right )^{3} - 4 \, a^{3} e \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}, x\right ) \]