\[ \int \frac {\sqrt {e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{13 d e \left (a +a \sin \left (d x +c \right )\right )^{4}}-\frac {10 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{117 a d e \left (a +a \sin \left (d x +c \right )\right )^{3}}-\frac {2 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{39 d e \left (a^{2}+a^{2} \sin \left (d x +c \right )\right )^{2}}-\frac {2 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{39 d e \left (a^{4}+a^{4} \sin \left (d x +c \right )\right )}-\frac {2 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{39 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^4,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 )^{4} e^{\frac {1}{2}} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} + 8 i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 4 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} + {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} + 4 i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 4 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} - 8 i \, \sqrt {2} e^{\frac {1}{2}}\right )} \sin \left (d x + c\right ) - 8 i \, \sqrt {2} e^{\frac {1}{2}}\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 )^{4} e^{\frac {1}{2}} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 8 i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} + 4 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} + {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 4 i \, \sqrt {2} \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} + 4 i \, \sqrt {2} \cos \left (d x + c\right ) e^{\frac {1}{2}} + 8 i \, \sqrt {2} e^{\frac {1}{2}}\right )} \sin \left (d x + c\right ) + 8 i \, \sqrt {2} e^{\frac {1}{2}}\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 (3 \, \cos \left (d x + c\right )^{4} e^{\frac {1}{2}} + 12 \, \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 14 \, \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 32 \, \cos \left (d x + c\right ) e^{\frac {1}{2}} + {\left (3 \, \cos \left (d x + c\right )^{3} e^{\frac {1}{2}} - 9 \, \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 23 \, \cos \left (d x + c\right ) e^{\frac {1}{2}} + 9 \, e^{\frac {1}{2}}\right )} \sin \left (d x + c\right ) - 9 \, e^{\frac {1}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{117 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} - 3 \, a^{4} d \cos \left (d x + c\right )^{3} - 8 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + 8 \, a^{4} d - {\left (a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d \cos \left (d x + c\right ) - 8 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )}}{a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )}, x\right ) \]