\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx \]
Optimal antiderivative \[ \frac {\left (B +8 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}+\frac {\left (3 A +4 B +17 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{42 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}+\frac {\left (15 A -B -83 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{210 a^{4} d \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {\left (B +8 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{10 a^{4} d \left (1+\cos \left (d x +c \right )\right )}-\frac {\left (A -B +C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{7 d \left (a +a \cos \left (d x +c \right )\right )^{4}}+\frac {\left (5 A +2 B -9 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{35 a d \left (a +a \cos \left (d x +c \right )\right )^{3}} \]
command
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2)/(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 (B + 8 \, C\right )} \cos \left (d x + c\right )^{3} - {\left (15 \, A - 64 \, B - 587 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (60 \, A - 53 \, B - 724 \, C\right )} \cos \left (d x + c\right ) - 15 \, A - 20 \, B + 335 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 5 \, {\left (\sqrt {2} {\left (3 i \, A + 4 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (3 i \, A + 4 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (3 i \, A + 4 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (3 i \, A + 4 i \, B + 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (3 i \, A + 4 i \, B + 17 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 (-3 i \, A - 4 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-3 i \, A - 4 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-3 i \, A - 4 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-3 i \, A - 4 i \, B - 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-3 i \, A - 4 i \, B - 17 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 (-i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-i \, B - 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-i \, B - 8 i \, 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 \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (i \, B + 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (i \, B + 8 i \, 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 )^{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
\[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {\cos \left (d x + c\right )}}{a^{4} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right )^{3} \sec \left (d x + c\right ) + a^{4} \cos \left (d x + c\right )^{3}}, x\right ) \]