Does the integral $\int _0^\infty \frac 1{1+(x \cos x)^2}\,dx$ converge?

Question

Question

$$\int _0^\infty \frac 1{1+(x \cos x)^2}dx$$ Does this converge?

Thoughts

Tried bounding it with $\frac 1{\cos^2x}$ for $x>1$ but not much success.