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$\large I_{n} = \int_{0}^{\frac{\pi}{4}} \tan ^{n}x \, dx$

We define the integral as above. And define the relation $f(n) = \dfrac{1}{I_{n-2} - I_{n+2}}$. Evaluate the summation below.

$\large 2\sum_{r=2}^{10} f(r)$

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