\[ \int \frac {A+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A +C \right ) \sin \left (d x +c \right )}{2 d \cos \left (d x +c \right )^{\frac {7}{2}} \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\left (8 A +19 C \right ) \arcsinh \left (\frac {\sqrt {a}\, \tan \left (d x +c \right )}{\sqrt {a +a \sec \left (d x +c \right )}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{4 a^{\frac {3}{2}} d}-\frac {\left (5 A +13 C \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {2}}{2 \sqrt {a +a \sec \left (d x +c \right )}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {2}}{4 a^{\frac {3}{2}} d}+\frac {\left (A +2 C \right ) \sin \left (d x +c \right )}{2 a d \cos \left (d x +c \right )^{\frac {5}{2}} \sqrt {a +a \sec \left (d x +c \right )}}-\frac {\left (2 A +7 C \right ) \sin \left (d x +c \right )}{4 a d \cos \left (d x +c \right )^{\frac {3}{2}} \sqrt {a +a \sec \left (d x +c \right )}} \]
command
integrate((A+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+a*sec(d*x+c))^(3/2),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 \[ \text {Timed out} \]_____________________________________________________