\[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx \]
Optimal antiderivative \[ -\frac {16 a^{2} \sec \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}-\frac {2 a \sec \left (d x +c \right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{3 d}+\frac {64 a^{3} \sec \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}}{3 d} \]
command
integrate(sec(d*x+c)^2*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {4 \, \sqrt {2} {\left (a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {3 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}\right )} \sqrt {a}}{3 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________