\[ \int (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \]
Optimal antiderivative \[ \frac {5 a^{\frac {5}{2}} \left (5 A +8 C \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right )}{8 d}-\frac {a^{3} \left (49 A -24 C \right ) \sin \left (d x +c \right )}{24 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {5 a A \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{12 d}+\frac {A \left (a +a \cos \left (d x +c \right )\right )^{\frac {5}{2}} \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{3 d}+\frac {a^{2} \left (31 A +24 C \right ) \sqrt {a +a \cos \left (d x +c \right )}\, \tan \left (d x +c \right )}{24 d} \]
command
integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^4,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 \[ \text {Timed out} \]_____________________________________________________