Happy nightmare

Calculus Level 5

\[I =\displaystyle\int \limits^{\infty }_{0}\displaystyle\int \limits^{\infty }_{0}\ln\left( \frac{x^{2}+2}{x^{2}+4} \right) \cos \left( xy\right) \, dx dy = \dfrac {-\pi}{A}\ln (A)\]

Find \(A\).

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