\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx \]
Optimal antiderivative \[ -\frac {\left (176 A -57 B +8 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}+\frac {\left (339 A -108 B +17 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{42 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d}-\frac {\left (43 A -15 B +C \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{42 a^{4} d \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {\left (176 A -57 B +8 C \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{30 a^{4} d \left (1+\cos \left (d x +c \right )\right )}-\frac {\left (A -B +C \right ) \left (\cos ^{\frac {9}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7 d \left (a +a \cos \left (d x +c \right )\right )^{4}}-\frac {\left (13 A -6 B -C \right ) \left (\cos ^{\frac {7}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35 a d \left (a +a \cos \left (d x +c \right )\right )^{3}}+\frac {\left (339 A -108 B +17 C \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{42 a^{4} d} \]
command
integrate(cos(d*x+c)^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^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 (140 \, A \cos \left (d x + c\right )^{4} + 7 \, {\left (368 \, A - 111 \, B + 24 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (6259 \, A - 1968 \, B + 337 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (5548 \, A - 1761 \, B + 284 \, C\right )} \cos \left (d x + c\right ) + 1695 \, A - 540 \, B + 85 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, {\left (\sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (339 i \, A - 108 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 (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-339 i \, A + 108 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 (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (176 i \, A - 57 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 (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-176 i \, A + 57 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
\[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) \sec \left (d x + c\right ) + A \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{a^{4} \sec \left (d x + c\right )^{4} + 4 \, a^{4} \sec \left (d x + c\right )^{3} + 6 \, a^{4} \sec \left (d x + c\right )^{2} + 4 \, a^{4} \sec \left (d x + c\right ) + a^{4}}, x\right ) \]