\[ \int \frac {\sqrt {a+a \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {a \left (8 B +C \right ) \sin \left (d x +c \right )}{24 d \sec \left (d x +c \right )^{\frac {5}{2}} \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a \left (48 A +40 B +35 C \right ) \sin \left (d x +c \right )}{96 d \sec \left (d x +c \right )^{\frac {3}{2}} \sqrt {a +a \cos \left (d x +c \right )}}+\frac {C \sin \left (d x +c \right ) \sqrt {a +a \cos \left (d x +c \right )}}{4 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {a \left (48 A +40 B +35 C \right ) \sin \left (d x +c \right )}{64 d \sqrt {a +a \cos \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )}}+\frac {\left (48 A +40 B +35 C \right ) \arcsin \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right ) \sqrt {a}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{64 d} \]
command
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*(a+a*cos(d*x+c))^(1/2)/sec(d*x+c)^(3/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \text {output too large to display} \]
Maxima 5.44 via sagemath 9.3 output \[ \text {Timed out} \]_____________________________________________________