\[ \int \frac {1}{(2 \cos (c+d x)+3 \sin (c+d x))^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {6 \,13^{\frac {1}{4}} \sqrt {\frac {\cos \left (c +d x -\arctan \left (\frac {3}{2}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (\frac {3}{2}\right )}{2}\right ), \sqrt {2}\right )}{845 \cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (\frac {3}{2}\right )}{2}\right ) d}-\frac {2 \left (3 \cos \left (d x +c \right )-2 \sin \left (d x +c \right )\right )}{65 d \left (2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {6 \left (3 \cos \left (d x +c \right )-2 \sin \left (d x +c \right )\right )}{845 d \sqrt {2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )}} \]
command
integrate(1/(2*cos(d*x+c)+3*sin(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {3 \, \sqrt {3 i + 2} {\left (46 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 9 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 3 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) - 54 i \, \sqrt {2} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (\frac {48}{13} i + \frac {20}{13}, 0, {\rm weierstrassPInverse}\left (\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {-3 i + 2} {\left (-46 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 9 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 3 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) + 54 i \, \sqrt {2} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-\frac {48}{13} i + \frac {20}{13}, 0, {\rm weierstrassPInverse}\left (-\frac {48}{13} i + \frac {20}{13}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (27 \, \cos \left (d x + c\right )^{3} + 2 \, {\left (69 \, \cos \left (d x + c\right )^{2} - 40\right )} \sin \left (d x + c\right ) + 48 \, \cos \left (d x + c\right )\right )} \sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}{845 \, {\left (46 \, d \cos \left (d x + c\right )^{3} - 54 \, d \cos \left (d x + c\right ) - 9 \, {\left (d \cos \left (d x + c\right )^{2} + 3 \, d\right )} \sin \left (d x + c\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}{119 \, \cos \left (d x + c\right )^{4} - 54 \, \cos \left (d x + c\right )^{2} + 24 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 81}, x\right ) \]