\[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {154 a^{4} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{15 d \,e^{3}}+\frac {4 a^{7} \left (e \cos \left (d x +c \right )\right )^{\frac {11}{2}}}{d \,e^{7} \left (a -a \sin \left (d x +c \right )\right )^{3}}+\frac {44 a^{8} \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{3 d \,e^{5} \left (a^{4}-a^{4} \sin \left (d x +c \right )\right )}-\frac {154 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{2} \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {231 \, {\left (-i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) + i \, \sqrt {2} a^{4} \sin \left (d x + c\right ) - i \, \sqrt {2} a^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, {\left (i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) - i \, \sqrt {2} a^{4} \sin \left (d x + c\right ) + i \, \sqrt {2} a^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, a^{4} \cos \left (d x + c\right )^{3} + 20 \, a^{4} \cos \left (d x + c\right )^{2} + 137 \, a^{4} \cos \left (d x + c\right ) + 120 \, a^{4} + {\left (3 \, a^{4} \cos \left (d x + c\right )^{2} - 17 \, a^{4} \cos \left (d x + c\right ) + 120 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{15 \, {\left (d \cos \left (d x + c\right ) e^{\frac {3}{2}} - d e^{\frac {3}{2}} \sin \left (d x + c\right ) + d e^{\frac {3}{2}}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]