I thought I was a good math student, but for some reason I am getting tripped up on complex analysis. Can someone please explain what I am doing wrong when I try to solve the following problem?

Problem: Given the function f(z)=\frac{z-a}{z+a}, where a\in\mathbb{C}, compute f^{(n)}(0) the "smart" way - using the Cauchy Integral Formula.

Attempted Solution:
If a=0, the problem is trivial, so assume a not zero. Cauchy's integral formula gives us that if C is any "nice" curve enclosing 0 in the complex plane, then
f^{(n)}(0)=\cfrac{n!}{2\pi i}\int_C{\cfrac{f(z)}{z^{n+1}}dz}
Let's take C to be a counter-clockwise circle about 0, of small enough radius that the singularity z=-a is excluded from the interior (so that Cauchy's integral formula applies. Now, we want to compute the integral
I=\int_C{\cfrac{z-a}{(z+a)z^{n+1}}dz}
Immediately using the parametrization z=re^{i\theta} doesn't appear to be fruitful here, but the substitution z=1/u yields:
I=-\int_{C'}{\cfrac{1/u-a}{1/u+a}u^{n-1}du}
=-\int_{C'}{\cfrac{1-ua}{1+ua}u^{n-1}du
where C' is now a circle of radius r'>1/|a|. Hence, this curve contains a singularity, u=-1/a. Where do I go from here? Any better ideas? (My book says to use Cauchy integral formula, so I'd appreciate solutions which somehow use that result.)