\[\int\limits_{0}^{\pi/2} \left( \frac{4J_{2}-16J_{4}+36J_{6}-64J_{8}+\ldots}{J_{1}-9J_{3}+25J_{5}-49J_{7}+\ldots}\right)^{2/3}\,\mathrm dx~=~?\]

**Details and Clarifications :**

- \(J_{n}\) denotes the Bessel function of the first kind.

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