\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{2 d \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {11 \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {2}}{2 \sqrt {a +a \sec \left (d x +c \right )}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {2}}{4 a^{\frac {3}{2}} d}-\frac {19 \sin \left (d x +c \right )}{6 a d \sqrt {\cos \left (d x +c \right )}\, \sqrt {a +a \sec \left (d x +c \right )}}+\frac {7 \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{6 a d \sqrt {a +a \sec \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^(3/2)/(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 \[ \text {Timed out} \]_____________________________________________________