\[ \int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx \]
Optimal antiderivative \[ -\frac {11 a^{5} \cos \left (d x +c \right )}{64 d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {11 a^{2} \left (\sec ^{5}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{140 d}+\frac {11 a \left (\sec ^{7}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{126 d}+\frac {\left (\sec ^{9}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{9 d}-\frac {11 a^{\frac {7}{2}} \arctanh \left (\frac {\cos \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (d x +c \right )}}\right ) \sqrt {2}}{128 d}+\frac {11 a^{4} \sec \left (d x +c \right )}{48 d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {11 a^{3} \left (\sec ^{3}\left (d x +c \right )\right ) \sqrt {a +a \sin \left (d x +c \right )}}{120 d} \]
command
integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} a^{\frac {7}{2}} {\left (\frac {630 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {4 \, {\left (1575 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 420 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 189 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}} - 3465 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 3465 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{80640 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________