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