\[ \int (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \, dx \]
Optimal antiderivative \[ \frac {11 a^{\frac {3}{2}} \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}}{\sqrt {a +a \cos \left (d x +c \right )}}\right )}{8 d}+\frac {11 a^{2} \tan \left (d x +c \right )}{8 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {11 a^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{12 d \sqrt {a +a \cos \left (d x +c \right )}}+\frac {a^{2} \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{3 d \sqrt {a +a \cos \left (d x +c \right )}} \]
command
integrate((a+a*cos(d*x+c))^(3/2)*sec(d*x+c)^4,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 \[ \text {Timed out} \]_____________________________________________________