\[I=\displaystyle\int_{0}^{\infty}\dfrac{\log\left(\frac{1+x^{4+\sqrt{15\vphantom{\large A}}}}{1+x^{2+\sqrt{3\vphantom{\large A}}}}\right)}{\left(1+x^2\right)\log x}\mathrm dx=\dfrac{\pi}{N^2}\left(N+\sqrt{O}\sqrt{P-\sqrt{Q}}\right)\].

Then Find \(N+O+P+Q\)

1)P,Q are co-prime

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