Does the following integral $$\int_{-1}^{1} \frac{\ln (\sqrt{3} x +2)}{\sqrt{1-x^{2}} (\sqrt{3} x + 2)^{n}}\ dx, \; \; n \in \mathbb{N}$$
have a nice closed form? Basically I cannot tackle it in any direction. Symmetry is useless . Applyig parts , well , gets things worse than they actually are. Could complex analysis help us here? That is, integrating around a dog bone contour ... I highly doubt it but it is just a thought.
Any help on this one?