\[ \int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 B \arctanh \left (\frac {\sin \left (x \right ) \sqrt {a}}{\sqrt {a +a \cos \left (x \right )}}\right )}{a^{\frac {5}{2}}}+\frac {\left (A -B \right ) \sin \left (x \right )}{4 \left (a +a \cos \left (x \right )\right )^{\frac {5}{2}}}+\frac {\left (3 A -11 B \right ) \sin \left (x \right )}{16 a \left (a +a \cos \left (x \right )\right )^{\frac {3}{2}}}+\frac {\left (3 A -43 B \right ) \arctanh \left (\frac {\sin \left (x \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \cos \left (x \right )}}\right ) \sqrt {2}}{32 a^{\frac {5}{2}}} \]
command
integrate((A+B*sec(x))/(a+a*cos(x))^(5/2),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 \[ \text {Timed out} \]_____________________________________________________