\[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A +C \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \cos \left (d x +c \right )}}\right ) \sqrt {2}}{d \sqrt {a}}+\frac {2 \left (15 A +14 C \right ) \sin \left (d x +c \right )}{15 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {2 C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d \sqrt {a +a \cos \left (d x +c \right )}}-\frac {2 C \sin \left (d x +c \right ) \sqrt {a +a \cos \left (d x +c \right )}}{15 a d} \]
command
integrate(cos(d*x+c)*(A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^(1/2),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 \[ \text {Timed out} \]_____________________________________________________