Is there an integral that proves \pi > 333/106\pi > 333/106?

The following integral,

is clearly positive, which proves that $\pi < 22/7$.

Is there a similar integral which proves $\pi > 333/106$?

$$\int_0^1 \frac{x^5(1-x)^6(197+462x^2)}{530(1+x^2)}\:dx= \pi -\frac{333}{106}\int_0^1 \frac{x^5(1-x)^6(197+462x^2)}{530(1+x^2)}\:dx= \pi -\frac{333}{106}$$
• Also you can refer to S.K. Lucas Integral proofs that $$355/113 > \pi355/113 > \pi$$, Gazette, Aust. Math. Soc. 32 (2005), 263-266.