I tried substituting $x=e^{2t}$ and rearranging the expression, but could not proceed any further. Apparently it breaks down into the difference of two separate integrals, both of which I don't believe can be computed by standard integration methods.
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Please see MathJax tutorial – Тyma Gaidash Jun 02 '23 at 12:05
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1Check out the answer at Integral ${\frac{1}{\pi^2}} \int_{0}^{\infty}\frac{{(\ln{x}})^2}{\sqrt{x}{(1-x)^2}} \mathrm d x$ – Sarvesh Ravichandran Iyer Jun 02 '23 at 12:08