\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{33 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d \sqrt {e \cos \left (d x +c \right )}}-\frac {2 \sqrt {e \cos \left (d x +c \right )}}{15 d e \left (a +a \sin \left (d x +c \right )\right )^{4}}-\frac {14 \sqrt {e \cos \left (d x +c \right )}}{165 a d e \left (a +a \sin \left (d x +c \right )\right )^{3}}-\frac {2 \sqrt {e \cos \left (d x +c \right )}}{33 d e \left (a^{2}+a^{2} \sin \left (d x +c \right )\right )^{2}}-\frac {2 \sqrt {e \cos \left (d x +c \right )}}{33 d e \left (a^{4}+a^{4} \sin \left (d x +c \right )\right )} \]
command
integrate(1/(a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {5 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{4} + 8 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 4 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) - 8 i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{4} - 8 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 4 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) + 8 i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (20 \, \cos \left (d x + c\right )^{2} + {\left (5 \, \cos \left (d x + c\right )^{2} - 37\right )} \sin \left (d x + c\right ) - 48\right )} \sqrt {\cos \left (d x + c\right )}}{165 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} e^{\frac {1}{2}} - 8 \, a^{4} d \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} + 8 \, a^{4} d e^{\frac {1}{2}} - 4 \, {\left (a^{4} d \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 2 \, a^{4} d e^{\frac {1}{2}}\right )} \sin \left (d x + c\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )}}{a^{4} e \cos \left (d x + c\right )^{5} - 8 \, a^{4} e \cos \left (d x + c\right )^{3} + 8 \, a^{4} e \cos \left (d x + c\right ) - 4 \, {\left (a^{4} e \cos \left (d x + c\right )^{3} - 2 \, a^{4} e \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}, x\right ) \]