\[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (3 A -2 B \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right )}{a^{\frac {3}{2}} d}+\frac {\left (9 A -5 B +C \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \cos \left (d x +c \right )}}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} d}-\frac {\left (A -B +C \right ) \tan \left (d x +c \right )}{2 d \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\left (3 A -B +C \right ) \tan \left (d x +c \right )}{2 a d \sqrt {a +a \cos \left (d x +c \right )}} \]
command
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+a*cos(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 \[ \text {Timed out} \]_____________________________________________________