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