\[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -B \right ) \left (\sec ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{2 d \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (39 A -35 B \right ) \left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{30 a d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {\left (9 A -5 B \right ) \left (\sec ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{10 a d \sqrt {a +a \cos \left (d x +c \right )}}-\frac {\left (15 A -11 B \right ) \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {\cos \left (d x +c \right )}\, \sqrt {a +a \cos \left (d x +c \right )}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {2}}{4 a^{\frac {3}{2}} d}+\frac {\left (147 A -95 B \right ) \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{30 a d \sqrt {a +a \cos \left (d x +c \right )}} \]
command
Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2),x]
Mathematica 13.1 output
\[ \frac {\cos ^3\left (\frac {1}{2} (c+d x)\right ) \left (-60 i (15 A-11 B) e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )+(264 A-120 B+(393 A-205 B) \cos (c+d x)+24 (9 A-5 B) \cos (2 (c+d x))+147 A \cos (3 (c+d x))-95 B \cos (3 (c+d x))) \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{60 d (a (1+\cos (c+d x)))^{3/2}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________