\[ \mathfrak{E} = \large \displaystyle \int_{1/2}^{1} \dfrac{2x \sin(2 \tan(x^2 ))}{\sin^2 (\tan(x^2)) \cos^2 (x^2)} \, dx \]

Find \( \left \lfloor 1000 \mathfrak{E} \right \rfloor \).

**Details and assumptions:**

After you have integrated and are ready to put in the limits of integration, of course by hand, then you may use a calculator to compute the final answer.

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