Closed form for $\int_0^1\frac{x}{\ln(x+1)(x^3+3x+3)}dx$

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How can I evaluate the closed form of the following integral: $$\int_0^1\frac{x}{\ln(x+1)(x^3+3x+3)}dx$$


According to Wolfram Alpha, the numerical value of this integral is close to 0.2673, but it doesn't show up any closed form.

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