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Strategy for Improper Integrals Related to the Beta Function 2

I am looking for the solution of the following integral $$\int_0^1 y^k \log\left(1+\left(\frac y{1-y}\right)^a\right)dy,\quad a>0 $$ I really appreciate it if any one can help.
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Strategy for Improper Integrals Related to the Beta Function

I'm interested in whether or not integrals of the form $I (\alpha , \beta , k) = \displaystyle\int_0^1 \log^k \left(\frac{x}{1-x}\right) x^{\alpha -1} (1-x)^{\beta -1 }dx,$ with $k\in \mathbb{N}, ...