Long intervals of consecutive composite values of polynomials

Let $f\in\mathbb Q[t]$ be an irreducible polynomial of positive degree with positive leading coefficient, taking integer values at every integer. We prove that there is a constant $c_f>0$ such that, for every sufficiently large real $X$, the interval $(X/2,X]$ contains at least $c_f\log X\log\log X$ consecutive integers $n$ for which all values $f(n)$ are composite. The exponent of $\log\log X$ is independent of the degree of $f$. This improves the previous result of Ford and Gabdullin by replacing their small fixed power of $\log\log X$ with the full factor $\log\log X$. In particular, the result applies to $f(t)=t^2+1$.