\[ \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {4 a b \left (3 A +5 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (7 b^{2} \left (A +3 C \right )+a^{2} \left (5 A +7 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {8 a A b \sin \left (d x +c \right )}{35 d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (4 A \,b^{2}+a^{2} \left (5 A +7 C \right )\right ) \sin \left (d x +c \right )}{21 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 A \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{7 d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {4 a b \left (3 A +5 C \right ) \sin \left (d x +c \right )}{5 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-42 i \, \sqrt {2} {\left (3 \, A + 5 \, C\right )} a b \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 42 i \, \sqrt {2} {\left (3 \, A + 5 \, C\right )} a b \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 5 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{2} + 7 i \, {\left (A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{2} - 7 i \, {\left (A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (42 \, {\left (3 \, A + 5 \, C\right )} a b \cos \left (d x + c\right )^{3} + 42 \, A a b \cos \left (d x + c\right ) + 15 \, A a^{2} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{2} + 7 \, A b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C b^{2} \cos \left (d x + c\right )^{4} + 2 \, C a b \cos \left (d x + c\right )^{3} + 2 \, A a b \cos \left (d x + c\right ) + A a^{2} + {\left (C a^{2} + A b^{2}\right )} \cos \left (d x + c\right )^{2}}{\cos \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]