\[ \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx \]
Optimal antiderivative \[ \frac {a^{\frac {3}{2}} \left (75 A +88 B \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right )}{64 d}+\frac {a^{2} \left (75 A +88 B \right ) \tan \left (d x +c \right )}{64 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a^{2} \left (75 A +88 B \right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )}{96 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a^{2} \left (9 A +8 B \right ) \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{24 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a A \left (\sec ^{3}\left (d x +c \right )\right ) \sqrt {a +a \cos \left (d x +c \right )}\, \tan \left (d x +c \right )}{4 d} \]
command
integrate((a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^5,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 \[ \text {Timed out} \]_____________________________________________________