Let $x = 2\tan{\theta}$, so $dx = 2\sec^2{\theta} d\theta$. When $x = -2$, $\theta = -\frac{\pi}{4}$, and when $x = 2$, $\theta = \frac{\pi}{4}$.
So, the integral becomes:
$$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{2\sec^2{\theta}}{4+4\tan^2{\theta}}
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