Integral with log and arctan

[Tolaso J Kos]

Has anyone considered working with integrals of the form:

$$\mathcal{J}_n=\int_0^1 \frac{\log^n x\arctan x}{1+x^2}\, {\rm d}x$$

I did today, but no headway. On the contrary I assume they are quite famous and they must be somewhere out there.

Helpful things

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• $\arctan x = \mathfrak{Im} \left [ \ln (1+ix) \right ]$
• $\displaystyle \frac{\arctan x}{1+x^2} = \sum_{m=1}^{\infty} (-1)^m \left ( \mathcal{H}_{2m} - \frac{1}{2} \mathcal{H}_m \right )x^{2m-1}$