Using contours to find where a function can be defined

Question

I need to find the largest open subset of $\mathbb{C}$ such that the complex holomorphic extension of $f:(0,\infty) \rightarrow \mathbb{R}$, given by $f(x) = \log \left( \dfrac{x}{x+1} \right)$, can be defined. I have been asked to do this using contour integration. I know that this is a holomorphic function where it is defined because we can use the logarithm sum property when we extend the real logarithm with the principal branch complex logarithm, giving us $\log \left( \dfrac{z}{z+1} \right) = \log(z) - \log(z+1)$ which are both holomorphic functions on their domains (please correct me if I am wrong with this). I know also that if a continuous function has an antiderivative, then we can use the fundamental theorem of calculus to see that integration over a closed contour over its domain is 0. I am therefore trying to show that the largest domain for this function (with branch cut on the real axis to the left of each logarithms singularities) is the entire complex plane without all negative purely real numbers by showing that a keyhole contour (in this case, an outer arc from $-\alpha$ to $\alpha$ followed by a line at angle $\alpha$ to an inner arc and so on) will give a zero integral because I am assuming that the antiderivative of the logarithm in the real case generalises to the complex case. However, when calculating the integral I am getting real logarithm terms for the arc integrals but they don't appear in the straight line contour integrals, so nothing cancels to give a zero integral before limits are taken. I therefore suspect I am wrong and that this won't actually be zero, but I can not see why. If it isn't, is there any suggestions for a better method to solve this?