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