\[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 a b \left (15 a^{2}+62 b^{2}\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{15 d \,e^{3}}+\frac {2 b \left (5 a^{2}+6 b^{2}\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +b \sin \left (d x +c \right )\right )}{5 d \,e^{3}}+\frac {2 a b \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +b \sin \left (d x +c \right )\right )^{2}}{d \,e^{3}}+\frac {2 \left (b +a \sin \left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{3}}{d e \sqrt {e \cos \left (d x +c \right )}}-\frac {2 \left (5 a^{4}+60 a^{2} b^{2}+12 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{2} \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+b*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
\[ -\frac {{\left (3 \, \sqrt {2} {\left (5 i \, a^{4} + 60 i \, a^{2} b^{2} + 12 i \, b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (-5 i \, a^{4} - 60 i \, a^{2} b^{2} - 12 i \, b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (20 \, a b^{3} \cos \left (d x + c\right )^{2} + 60 \, a^{3} b + 60 \, a b^{3} + 3 \, {\left (b^{4} \cos \left (d x + c\right )^{2} + 5 \, a^{4} + 30 \, a^{2} b^{2} + 5 \, b^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {3}{2}\right )}}{15 \, d \cos \left (d x + c\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{4} \cos \left (d x + c\right )^{4} + a^{4} + 6 \, a^{2} b^{2} + b^{4} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left (a b^{3} \cos \left (d x + c\right )^{2} - a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]