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### How to evaluate $\int_{0}^{\infty}\frac{(x^2-1)\ln{x}}{1+x^4}dx$?

How to evaluate the following integral $$I=\int_{0}^{\infty}\dfrac{(x^2-1)\ln{x}}{1+x^4}dx=\dfrac{\pi^2}{4\sqrt{2}}$$ without using residue or complex analysis methods?
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### How to compute $\sum_{k = 1}^{\infty} \frac{1}{k^2-\frac{1}{16}}$? [duplicate]
How do I compute $\sum_{k = 1}^{\infty} \frac{1}{k^2-\frac{1}{16}}$ ? Mathematica says the sum converges and it somewhat looks like the Basel problem, but so far I do not know how to approach it.