\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4} \, dx \]
Optimal antiderivative \[ -\frac {\left (8 A +B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}+\frac {\left (17 A +4 B +3 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{42 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}-\frac {\left (A -B +C \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7 d \left (a +a \cos \left (d x +c \right )\right )^{4}}-\frac {\left (9 A -2 B -5 C \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35 a d \left (a +a \cos \left (d x +c \right )\right )^{3}}-\frac {\left (83 A +B -15 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{210 a^{4} d \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {\left (8 A +B \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{10 a^{4} d \left (1+\cos \left (d x +c \right )\right )} \]
command
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2)/(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 (8 \, A + B\right )} \cos \left (d x + c\right )^{3} + {\left (337 \, A + 104 \, B + 15 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (284 \, A + 73 \, B + 60 \, C\right )} \cos \left (d x + c\right ) + 85 \, A + 20 \, B + 15 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, {\left (\sqrt {2} {\left (17 i \, A + 4 i \, B + 3 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (17 i \, A + 4 i \, B + 3 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (17 i \, A + 4 i \, B + 3 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (17 i \, A + 4 i \, B + 3 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (17 i \, A + 4 i \, B + 3 i \, 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 (-17 i \, A - 4 i \, B - 3 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-17 i \, A - 4 i \, B - 3 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-17 i \, A - 4 i \, B - 3 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-17 i \, A - 4 i \, B - 3 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-17 i \, A - 4 i \, B - 3 i \, 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 (8 i \, A + i \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (8 i \, A + i \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (8 i \, A + i \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (8 i \, A + i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (8 i \, A + i \, B\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 (-8 i \, A - i \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-8 i \, A - i \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-8 i \, A - i \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-8 i \, A - i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-8 i \, A - i \, B\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 )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {\cos \left (d x + c\right )}}{a^{4} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right ) \sec \left (d x + c\right ) + a^{4} \cos \left (d x + c\right )}, x\right ) \]