\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {4 a^{3} \left (5 A +7 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{3} \left (105 A +143 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {8 a^{3} \left (35 A +44 C \right ) \sin \left (d x +c \right )}{385 d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 a^{3} \left (105 A +143 C \right ) \sin \left (d x +c \right )}{231 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 A \left (a +a \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{11 d \cos \left (d x +c \right )^{\frac {11}{2}}}+\frac {4 A \left (a^{2}+a^{2} \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{33 a d \cos \left (d x +c \right )^{\frac {9}{2}}}+\frac {2 \left (35 A +33 C \right ) \left (a^{3}+a^{3} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{231 d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {4 a^{3} \left (5 A +7 C \right ) \sin \left (d x +c \right )}{5 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (462 \, {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 10 \, {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (10 \, A + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (42 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, A a^{3} \cos \left (d x + c\right ) + 105 \, A a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{1155 \, d \cos \left (d x + c\right )^{6}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C a^{3} \cos \left (d x + c\right )^{5} + 3 \, C a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + {\left (3 \, A + C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, A a^{3} \cos \left (d x + c\right ) + A a^{3}}{\cos \left (d x + c\right )^{\frac {13}{2}}}, x\right ) \]