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A so-called complicate integral has a such a simple closed form, quite amazed me, but how to prove it, is an other story.

$$\int_{0}^{1}\mathrm dt{t-3t^3+t^5\over 1+t^4+t^8}\cdot \ln(-\ln t) dt=\color{red}{{\pi\over 3\sqrt{3}}}\cdot \color{blue}{\ln 2\over 2}$$

Does anyone know to how prove this integral?