\[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {7 a^{3} \cos \left (d x +c \right )}{16 d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\left (\sec ^{5}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{5 d}-\frac {7 a^{\frac {3}{2}} \arctanh \left (\frac {\cos \left (d x +c \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (d x +c \right )}}\right ) \sqrt {2}}{32 d}+\frac {7 a^{2} \sec \left (d x +c \right )}{12 d \sqrt {a +a \sin \left (d x +c \right )}}+\frac {7 a \left (\sec ^{3}\left (d x +c \right )\right ) \sqrt {a +a \sin \left (d x +c \right )}}{30 d} \]
command
integrate(sec(d*x+c)^6*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} a^{\frac {3}{2}} {\left (\frac {30 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {4 \, {\left (45 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - 105 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 105 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{960 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________