Calculate complex curve integral

Question

I need to calculate the integral:

$$ \int_{|z| = 2} \sin(2\cos(z) - 1)^5 \text{dz} $$

For now I only got:

$$ \int_0^1 \sin(\cos(2e^{2\pi it}) - 1)^5 \cdot 4\pi i e^{2\pi it} \text{dt} $$

and can't find any good way to proceed. Thank you for any further help!

Answer 1

The answer is zero since the integrand is an entire function and hence for any closed curve integrates to zero. I doubt the intent of the questioner is for you to actually calculate it. In fact, the point is quite the opposite.