Planning to integrate $\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$ using complex analysis

Question

This is just a plan-out.

I want to evaluate:

$$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$

Using a keyhole contour a semi-circle, with base at the x-axis.

First I must pick a branch.

Since the logarithm is multivalued in that,

$$\log(e^{i2\pi}) = \log(e^{(2ki\pi) + i2\pi})$$

We consider: $f(z) = \displaystyle \frac{\log^2(z)}{z^2 + 1}$

But how should I pick a branch.

(1) What is a "Branch?"

(2) Should I pick a place where the $\log$ is continous. That is everywhere except $z=0$

So then my contour will be half-washer shape. with the opening in the middle: Half a Washer

Thanks