\[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (24 a^{2} C +7 b^{2} \left (9 A +7 C \right )\right ) \sin \left (d x +c \right )}{315 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 a \left (63 A \,b^{2}+8 a^{2} C +45 b^{2} C \right ) \sin \left (d x +c \right )}{63 d \sqrt {\sec \left (d x +c \right )}}+\frac {4 a C \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{21 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 C \left (a +b \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{9 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 b \left (9 a^{2} \left (5 A +3 C \right )+b^{2} \left (9 A +7 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a \left (7 a^{2} \left (3 A +C \right )+3 b^{2} \left (7 A +5 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {15 \, \sqrt {2} {\left (7 i \, {\left (3 \, A + C\right )} a^{3} + 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-7 i \, {\left (3 \, A + C\right )} a^{3} - 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b - i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b + i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, C b^{3} \cos \left (d x + c\right )^{4} + 135 \, C a b^{2} \cos \left (d x + c\right )^{3} + 7 \, {\left (27 \, C a^{2} b + {\left (9 \, A + 7 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (7 \, C a^{3} + 3 \, {\left (7 \, A + 5 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{5} + 3 \, C a b^{2} \cos \left (d x + c\right )^{4} + 3 \, A a^{2} b \cos \left (d x + c\right ) + A a^{3} + {\left (3 \, C a^{2} b + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]