To use Cauchy's Integral Formula, as you title this thread, I would note that then use use "partial fractions" to write the integrand as a sum of three fractions and then do three separate integrals, using Cauchy's Integral Formula on the three separate integrals.
As FernandoRevilla implies, the answer depends upon the particular contour. In fact, because you have three different points in question, z= 0, z= i, and z= -i, and the value of each integral depends upon whether the point is or is not in the contour, there will be answers. For example, if the contour is such that NONE of the three points are in it, the integral in question is 0. If z= 0 is in the integral but not -i or i, you will get one answer, if z= i is in the integral but not 0 or -i, another, if both 0 and i are in the contour but not -i, yet another, etc. (Since the radius is given as 2, you cannot have all three points inside the contour so there are really at most 7 different answers.)