Evaluate $\int_{-\infty}^\infty \frac{1}{x^4 + 7x^2 + 6} dx$ using Fourier analysis

Question

By using Fourier transform methods compute the integral $$\int_{-\infty}^\infty \frac{1}{x^4 + 7x^2 + 6} \ dx.$$

I know that I should use the inversion formula and evaluate it in some point, but I can't figure out how to do this.