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Mathematica can do nothing with this expectation in general:

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

However, we have this:

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:

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

Mathematica can do nothing with this expectation in general:

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

However, we have this:

and this:

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:

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

Mathematica can do nothing this expectation in general:

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

However, we have this:

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:

Mathematica can do nothing with this expectation in general:

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

However, we have this:

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:

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

Mathematica can do nothing this expectation in general:

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

However, we have this:

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

Mathematica can do nothing this expectation in general:

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

However, we have this:

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: