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I have come across the following inequality in my studies

$$\frac{\text{arccot}2\sqrt{2}}{\pi\log\zeta(3)}-\frac{\log^2(1+e^{-\pi})}{\pi}>\frac{131e^2+422e-1151}{222e^2+279e-757},$$

where $\zeta(3)$ is Apéry's constant and $\log^2 x$ is the square of the natural logarithm of $x$. The RHS seems to be relatively close to the numerical value of the LHS, however I would like to know if there exists a way to prove this without relying on numerical computations.

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    $\begingroup$ If someone actually proves this in an answer I'm gonna eat my hat $\endgroup$ – Spine Feast Jan 16 '15 at 17:17
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    $\begingroup$ The OP is clearly trying to troll us all. $\endgroup$ – Jack D'Aurizio Jan 16 '15 at 17:27
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    $\begingroup$ Possibly relevant. Also see this. $\endgroup$ – Dejan Govc Jan 16 '15 at 17:35
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    $\begingroup$ It is a trolling. :) $\endgroup$ – Alex Silva Jan 16 '15 at 18:06
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    $\begingroup$ "I have come across the following inequality in my studies..." No, you haven't. $\endgroup$ – mjqxxxx Jan 16 '15 at 18:47

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