\[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b \cos \left (d x +c \right )-a \sin \left (d x +c \right )\right )}{5 \left (a^{2}+b^{2}\right ) d \left (a \cos \left (d x +c \right )+b \sin \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {6 \left (b \cos \left (d x +c \right )-a \sin \left (d x +c \right )\right )}{5 \left (a^{2}+b^{2}\right )^{2} d \sqrt {a \cos \left (d x +c \right )+b \sin \left (d x +c \right )}}-\frac {6 \sqrt {\frac {\cos \left (c +d x -\arctan \left (a , b\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (a , b\right )}{2}\right ), \sqrt {2}\right ) \sqrt {a \cos \left (d x +c \right )+b \sin \left (d x +c \right )}}{5 \cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\arctan \left (a , b\right )}{2}\right ) \left (a^{2}+b^{2}\right )^{2} d \sqrt {\frac {a \cos \left (d x +c \right )+b \sin \left (d x +c \right )}{\sqrt {a^{2}+b^{2}}}}} \]
command
integrate(1/(a*cos(d*x+c)+b*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 \, {\left (-3 i \, \sqrt {2} a b^{2} \cos \left (d x + c\right ) + \sqrt {2} {\left (-i \, a^{3} + 3 i \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (-i \, \sqrt {2} b^{3} + \sqrt {2} {\left (-3 i \, a^{2} b + i \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a - i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a^{2} + 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} + 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (3 i \, \sqrt {2} a b^{2} \cos \left (d x + c\right ) + \sqrt {2} {\left (i \, a^{3} - 3 i \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (i \, \sqrt {2} b^{3} + \sqrt {2} {\left (3 i \, a^{2} b - i \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a + i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a^{2} - 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a^{2} - 2 i \, a b - b^{2}\right )}}{a^{2} + b^{2}}, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (5 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right ) - {\left (a^{3} + 4 \, a b^{2} + 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}}{5 \, {\left ({\left (a^{7} - a^{5} b^{2} - 5 \, a^{3} b^{4} - 3 \, a b^{6}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d \cos \left (d x + c\right ) + {\left ({\left (3 \, a^{6} b + 5 \, a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}}{{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + b^{4} + 2 \, {\left (3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (a b^{3} \cos \left (d x + c\right ) + {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]