Mathematica can do nothing with this expectation in general: 

[![enter image description here][1]][1]

So, it is highly unlikely that this expectation can be expressed in terms of elementary, or even special, functions. 

However, we have this: 

[![enter image description here][2]][2]

Also, writing $\ln(1-x^a)=-\sum_{k=1}^\infty\dfrac{x^{ak}}k$, we see that the expectation in question is 
$$-\frac{\Gamma (\alpha+\beta )}
{\Gamma (\alpha)}\sum_{k=1}^\infty\frac{\Gamma (a k+\alpha )}{k\,\Gamma (a k+\alpha +\beta )}.$$ 
It is highly unlikely as well that the latter sum can be expressed in terms of elementary, or even special, functions: 

[![enter image description here][3]][3]

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Of course, one can compute this expectation numerically with any degree of accuracy. E.g., we have this: 

[![enter image description here][4]][4]


  [1]: https://i.sstatic.net/RCJOv.png
  [2]: https://i.sstatic.net/s1z4e.png
  [3]: https://i.sstatic.net/5yo5H.png
  [4]: https://i.sstatic.net/YN38s.png