\[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx \]
Optimal antiderivative \[ -\frac {8 a^{2} \left (\sec ^{3}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}+\frac {2 a \left (\sec ^{3}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{d} \]
command
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (3 \, a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{3 \, d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________