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How to calculate the following integral symbolically with Mathematica:

$$ I = \int_0^\infty \frac{\mathrm{d}x}{(x^2+a^2)(\ln^2x+\pi^2)} $$

where $a>0$ is a real number.

PS: from complex analysis, we can determine it:

$$ I = \frac{2\pi}{(4\ln^2a+\pi^2) \, a} - \frac{1}{a^2+1} $$

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