Complex Integral with Cauchy's Formula problem.

Question

I'm studying for my first complex analysis exam and I'm having problems understanding an exercise. So, I'm asked to calculate $\int_{|{z-1}|=3}\frac{dz}{z(z^2-4)e^z}$ using the Cauchy's formula. Here is what I did. Assuming that the question is asking me to calculate the intgral over the circumference with center in z=1 and radius 3, I noticed that inside that domain we have 3 singularities: z=0, z=2, z=-2. The idea was to "change" the domain of integration in three little circles centered in the singularities since in every other point the function is holomorphing and therefore the integral over every closed path is zero. Once i split the domain in this way all I have to do is apply Cauchy's formula over those three little circles. So I have $\int_{|{z-1}|=3}\frac{dz}{z(z^2-4)e^z}=\int_{|z-2|=\epsilon}\frac{dz}{z(z-2)(z+2)e^z} + \int_{|z|=\epsilon}\frac{dz}{z(z-2)(z+2)e^z} + \int_{|z+2|=\epsilon}\frac{dz}{z(z-2)(z+2)e^z} = 2\pi i(\frac{e^2}{8}+\frac{e^{-2}}{8}-\frac14)$. Now the problems are that I know that the answer has to be $-\frac{\pi i}{4e^2}$ and I don't know what I did wrong. I am quite sure I can do what I did for at least the singularities inside the domain z=0 and z=2, while I am not so sure how to treat the singularity over the frontier z=-2. Hope you guys can help me :).