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I ran across a curious integral that seems to be rather tough that some on the site may enjoy.

Show that $$\displaystyle \int_{0}^{1}\frac{\sqrt{1-x^{2}}}{1-x^{2}\sin^{2}(x)}dx = \frac{5\sqrt[5]{{\pi}^{8}}}{32\sqrt[5]{{\zeta(5)}^{9}}}$$

How in the world can $\zeta(5)$ be incorporated into this?. I tried series and several methods, but made no real progress. Any ideas?. Thanks very much.

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    This one is really tough!2011-11-23
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    Where did you see this relation?2011-11-23
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    I saw it on a site where someone had posted the solution but no method. At mymathforum.com. I checked this closed form against the numerical solution that Maple and Mathematica gave and it was exactly as posted. I do not know where the poster may have came up with it, but it appears to be correct.2011-11-23
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    I computed both sides to 20 places in Maple. It concluded they agree only to 8 places, and differ by about $2 \times 10^{-9}$.2011-11-24

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