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Evaluate: \[\int_0^{\infty} \frac{x^3\ln^2 x}{1+x^6}\,dx \]

If the result can be expressed as \(\displaystyle \frac{a\pi^b}{c\sqrt{d}}\), where ...

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The integral above has a closed form. Find this closed form.

Give your ...

Given \[2^{2014}\prod_{k=0}^{2014}\sin\left(x+\frac{k \pi}{2015}\right) = \sin\left(\frac{px}{q}\right).\] Find \(p+q\).

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