\[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx \]
Optimal antiderivative \[ \frac {a^{\frac {5}{2}} \left (1015 A +1132 B +1304 C \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right )}{512 d}+\frac {a \left (5 A +12 B \right ) \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (\sec ^{4}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{60 d}+\frac {A \left (a +a \cos \left (d x +c \right )\right )^{\frac {5}{2}} \left (\sec ^{5}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{6 d}+\frac {a^{3} \left (1015 A +1132 B +1304 C \right ) \tan \left (d x +c \right )}{512 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a^{3} \left (1015 A +1132 B +1304 C \right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )}{768 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a^{3} \left (545 A +628 B +680 C \right ) \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{960 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a^{2} \left (115 A +156 B +120 C \right ) \left (\sec ^{3}\left (d x +c \right )\right ) \sqrt {a +a \cos \left (d x +c \right )}\, \tan \left (d x +c \right )}{480 d} \]
command
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________