\[ \int \sec ^9(c+d x) (a+a \sin (c+d x))^{7/2} \, dx \]
Optimal antiderivative \[ \frac {21 a^{2} \left (\sec ^{4}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{256 d}+\frac {3 a \left (\sec ^{6}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{32 d}+\frac {\left (\sec ^{8}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{8 d}+\frac {315 a^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {2}}{4096 d}-\frac {315 a^{4}}{2048 d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {105 a^{3} \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {a +a \sin \left (d x +c \right )}}{1024 d} \]
command
integrate(sec(d*x+c)^9*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} a^{\frac {7}{2}} {\left (\frac {256}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {2 \, {\left (187 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 643 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 765 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 325 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}} - 315 \, \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 315 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{8192 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________