\[ \int \frac {\cos ^2(c+d x) (A+B \cos (c+d x))}{\sqrt {a+a \cos (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {\left (A -B \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \cos \left (d x +c \right )}}\right ) \sqrt {2}}{d \sqrt {a}}-\frac {4 \left (5 A -7 B \right ) \sin \left (d x +c \right )}{15 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {2 B \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {2 \left (5 A -B \right ) \sin \left (d x +c \right ) \sqrt {a +a \cos \left (d x +c \right )}}{15 a d} \]
command
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c))/(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 \[ \text {Timed out} \]_____________________________________________________