Improper Integrals Convergent or Divergent $\int_1^\infty \frac{2}{x(1+\cos^2(x^2+x+1))}dx$

Question

I have the integral: $$\int_1^\infty \frac{2}{x(1+\cos^2(x^2+x+1))}dx$$ I could not figure out how to represent the equation in the denominator, so I could apply the limit. I need only to find if it is convergent or divergent, I do not care about the value.

Answer 1

$$\frac2{x(1 + \cos^2(...))} \ge \frac2{x(1 + 1)} = \frac1x$$