\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx \]
Optimal antiderivative \[ \frac {\left (A +8 B -57 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}+\frac {\left (4 A +17 B -108 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{42 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}-\frac {\left (A +8 B -57 C \right ) \sin \left (d x +c \right )}{10 a^{4} d \sqrt {\cos \left (d x +c \right )}}+\frac {\left (13 A +29 B -141 C \right ) \sin \left (d x +c \right )}{210 a^{4} d \left (1+\cos \left (d x +c \right )\right )^{2} \sqrt {\cos \left (d x +c \right )}}+\frac {\left (4 A +17 B -108 C \right ) \sin \left (d x +c \right )}{42 a^{4} d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}}-\frac {\left (A -B +C \right ) \sin \left (d x +c \right )}{7 d \left (a +a \cos \left (d x +c \right )\right )^{4} \sqrt {\cos \left (d x +c \right )}}+\frac {\left (3 A +4 B -11 C \right ) \sin \left (d x +c \right )}{35 a d \left (a +a \cos \left (d x +c \right )\right )^{3} \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+a*sec(d*x+c))^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (21 \, {\left (A + 8 \, B - 57 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (64 \, A + 587 \, B - 4248 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (53 \, A + 724 \, B - 5421 \, C\right )} \cos \left (d x + c\right )^{2} - 5 \, {\left (4 \, A - 67 \, B + 564 \, C\right )} \cos \left (d x + c\right ) - 420 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 5 \, {\left (\sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, \sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, \sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, \sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, \sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, \sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, \sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, \sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, \sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, \sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )\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 \, {\left (\sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, \sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, \sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, \sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )\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 )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {\cos \left (d x + c\right )}}{a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right ) + a^{4} \cos \left (d x + c\right )^{4}}, x\right ) \]