How to compute this line integral?

Question

How do you compute this integral? $$\int_0^{2\pi} e^{R \cos t} \cos(\sin t+3t)dt \quad R>0.$$ My friend sent this to me the other day, but have not made much progress so far. Maybe using line integrals would be the easiest way to go about, since it looks like it will get messy very fast. He says that the answer should be $0$ which makes me think that this thing is symmetric over $\pi/2$, but not sure.

Answer 1

I did not obtain any formal result.

The center of symmetry is at $t=\pi$.

What I suspect (pure intuition) is that $$f(R)=\int_0^{2\pi} e^{R \cos (t)} \cos[3t+\sin (t)]\,dt$$ would increase "almost" exponentially since, close to $t=0$ the integrand is $$e^R\Big[1-\frac{1}{2} (R+16)\, t^2\Big]+O\left(t^4\right)$$

Just a few numbers (by numerical integration) $$\left( \begin{array}{cc} R & f(R) \\ 2 & 0.157363 \\ 3 & 1.688026 \\ 4 & 8.387231 \\ 5 & 31.62618 \\ 6 & 104.6882 \\ 7 & 322.8717 \\ 8 & 954.7903 \\ 9 & 2749.305 \\ 10 & 7777.683 \end{array} \right)$$