\[ \int \frac {(a+b \cos (c+d x))^3 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (7 a^{3} B +27 B a \,b^{2}+27 a^{2} b C +15 C \,b^{3}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (15 a^{2} b B +7 b^{3} B +5 a^{3} C +21 C a \,b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a^{2} \left (13 b B +9 a C \right ) \sin \left (d x +c \right )}{63 d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 a \left (7 B \,a^{2}+22 b^{2} B +27 a b C \right ) \sin \left (d x +c \right )}{45 d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (15 a^{2} b B +7 b^{3} B +5 a^{3} C +21 C a \,b^{2}\right ) \sin \left (d x +c \right )}{21 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 a B \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{9 d \cos \left (d x +c \right )^{\frac {9}{2}}}+\frac {2 \left (7 a^{3} B +27 B a \,b^{2}+27 a^{2} b C +15 C \,b^{3}\right ) \sin \left (d x +c \right )}{15 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+b*cos(d*x+c))^3*(B*cos(d*x+c)+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
\[ -\frac {15 \, \sqrt {2} {\left (5 i \, C a^{3} + 15 i \, B a^{2} b + 21 i \, C a b^{2} + 7 i \, B b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, C a^{3} - 15 i \, B a^{2} b - 21 i \, C a b^{2} - 7 i \, B b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (7 i \, B a^{3} + 27 i \, C a^{2} b + 27 i \, B a b^{2} + 15 i \, C b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-7 i \, B a^{3} - 27 i \, C a^{2} b - 27 i \, B a b^{2} - 15 i \, C b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (21 \, {\left (7 \, B a^{3} + 27 \, C a^{2} b + 27 \, B a b^{2} + 15 \, C b^{3}\right )} \cos \left (d x + c\right )^{4} + 35 \, B a^{3} + 15 \, {\left (5 \, C a^{3} + 15 \, B a^{2} b + 21 \, C a b^{2} + 7 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (7 \, B a^{3} + 27 \, C a^{2} b + 27 \, B a b^{2}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C b^{3} \cos \left (d x + c\right )^{4} + B a^{3} + {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (C a^{2} b + B a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]