I need help in solving the following problem in complex calculus. I need to calculate this real integral: $ \int_0^{2\pi} e^{\cos{\theta}}\cos(\sin\theta) \,\,\, d\theta $
I think that the best method is to do the substitution $z=re^{i\theta}$, from which we have $d\theta = \frac{dz}{iz}$, $\cos\theta=\frac{1}{2}(z+\frac{1}{z})$ and so on. The integral then becomes (without constants) $ \oint_{|z|=r} e^{\frac{z}{2}}e^{\frac{1}{2z}}\frac{z^4-6z^2+1}{z(z^2-1)} \,\,\,dz $
At this point I am unable to calculate the residue at the essential singularity in 0, because the manipulation of Laurent series rapidly becomes utterly complicated and I don't see any simple way to find the final coefficient of $1/z$. So, which is the way to handle this integral? Another substitution?