\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 a \left (192 A \,b^{3}+1053 a^{3} B +2171 B a \,b^{2}+a^{2} \left (2518 A b +3146 C b \right )\right ) \sin \left (d x +c \right )}{9009 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (192 A \,b^{4}+4004 a^{3} b B +3458 a \,b^{3} B +77 a^{4} \left (11 A +13 C \right )+11 a^{2} b^{2} \left (491 A +637 C \right )\right ) \sin \left (d x +c \right )}{6435 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 \left (48 A \,b^{2}+221 a b B +11 a^{2} \left (11 A +13 C \right )\right ) \left (a +b \sec \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{1287 d \sec \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 \left (8 A b +13 B a \right ) \left (a +b \sec \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{143 d \sec \left (d x +c \right )^{\frac {9}{2}}}+\frac {2 A \left (a +b \sec \left (d x +c \right )\right )^{4} \sin \left (d x +c \right )}{13 d \sec \left (d x +c \right )^{\frac {11}{2}}}+\frac {2 \left (45 a^{4} B +330 B \,a^{2} b^{2}+77 b^{4} B +44 a \,b^{3} \left (5 A +7 C \right )+20 a^{3} b \left (9 A +11 C \right )\right ) \sin \left (d x +c \right )}{231 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (364 a^{3} b B +468 a \,b^{3} B +39 b^{4} \left (3 A +5 C \right )+78 a^{2} b^{2} \left (7 A +9 C \right )+a^{4} \left (77 A +91 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 )}{195 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (45 a^{4} B +330 B \,a^{2} b^{2}+77 b^{4} B +44 a \,b^{3} \left (5 A +7 C \right )+20 a^{3} b \left (9 A +11 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 )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(13/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {195 \, \sqrt {2} {\left (45 i \, B a^{4} + 20 i \, {\left (9 \, A + 11 \, C\right )} a^{3} b + 330 i \, B a^{2} b^{2} + 44 i \, {\left (5 \, A + 7 \, C\right )} a b^{3} + 77 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 \, \sqrt {2} {\left (-45 i \, B a^{4} - 20 i \, {\left (9 \, A + 11 \, C\right )} a^{3} b - 330 i \, B a^{2} b^{2} - 44 i \, {\left (5 \, A + 7 \, C\right )} a b^{3} - 77 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-7 i \, {\left (11 \, A + 13 \, C\right )} a^{4} - 364 i \, B a^{3} b - 78 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b^{2} - 468 i \, B a b^{3} - 39 i \, {\left (3 \, A + 5 \, C\right )} b^{4}\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 ) + 231 \, \sqrt {2} {\left (7 i \, {\left (11 \, A + 13 \, C\right )} a^{4} + 364 i \, B a^{3} b + 78 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b^{2} + 468 i \, B a b^{3} + 39 i \, {\left (3 \, A + 5 \, C\right )} b^{4}\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 (3465 \, A a^{4} \cos \left (d x + c\right )^{6} + 4095 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{5} + 385 \, {\left ({\left (11 \, A + 13 \, C\right )} a^{4} + 52 \, B a^{3} b + 78 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + 585 \, {\left (9 \, B a^{4} + 4 \, {\left (9 \, A + 11 \, C\right )} a^{3} b + 66 \, B a^{2} b^{2} + 44 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (7 \, {\left (11 \, A + 13 \, C\right )} a^{4} + 364 \, B a^{3} b + 78 \, {\left (7 \, A + 9 \, C\right )} a^{2} b^{2} + 468 \, B a b^{3} + 117 \, A b^{4}\right )} \cos \left (d x + c\right )^{2} + 195 \, {\left (45 \, B a^{4} + 20 \, {\left (9 \, A + 11 \, C\right )} a^{3} b + 330 \, B a^{2} b^{2} + 44 \, {\left (5 \, A + 7 \, C\right )} a b^{3} + 77 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45045 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C b^{4} \sec \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \sec \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \sec \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \sec \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {13}{2}}}, x\right ) \]