\[ \int \sin ^n(c+d x) (a+a \sin (c+d x))^{-2-n} (-1-n-(-2-n) \sin (c+d x)) \, dx \]
Optimal antiderivative \[ -\frac {\cos \left (d x +c \right ) \left (\sin ^{1+n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{-2-n}}{d} \]
command
integrate(sin(d*x+c)^n*(a+a*sin(d*x+c))^(-2-n)*(-1-n-(-2-n)*sin(d*x+c)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________