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$$\int_{0}^{1} \frac{\arctan(x)}{1 + x^2} \cdot \operatorname{Li}_{2}\left(\frac{1 - 4x + 6x^2 - 4x^3 + x^4}{1 + 4x + 6x^2 + 4x^3 + x^4}\right) \, dx$$