\[ \int \frac {(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b +a \sin \left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{3}}{7 d e \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {2 \left (a +b \sin \left (d x +c \right )\right )^{2} \left (a b -\left (5 a^{2}-6 b^{2}\right ) \sin \left (d x +c \right )\right )}{21 d \,e^{3} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 \left (5 a^{4}-12 a^{2} b^{2}+12 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{4} \sqrt {e \cos \left (d x +c \right )}}+\frac {10 a b \left (a^{2}-2 b^{2}\right ) \sqrt {e \cos \left (d x +c \right )}}{21 d \,e^{5}}+\frac {2 b \left (5 a^{2}-6 b^{2}\right ) \left (a +b \sin \left (d x +c \right )\right ) \sqrt {e \cos \left (d x +c \right )}}{21 d \,e^{5}} \]
command
integrate((a+b*sin(d*x+c))^4/(e*cos(d*x+c))^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (\sqrt {2} {\left (-5 i \, a^{4} + 12 i \, a^{2} b^{2} - 12 i \, b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (5 i \, a^{4} - 12 i \, a^{2} b^{2} + 12 i \, b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (28 \, a b^{3} \cos \left (d x + c\right )^{2} - 12 \, a^{3} b - 12 \, a b^{3} - {\left (3 \, a^{4} + 18 \, a^{2} b^{2} + 3 \, b^{4} + {\left (5 \, a^{4} - 12 \, a^{2} b^{2} - 9 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}\right )} e^{\left (-\frac {9}{2}\right )}}{21 \, d \cos \left (d x + c\right )^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{4} \cos \left (d x + c\right )^{4} + a^{4} + 6 \, a^{2} b^{2} + b^{4} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left (a b^{3} \cos \left (d x + c\right )^{2} - a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{5} \cos \left (d x + c\right )^{5}}, x\right ) \]