I am trying to compute
$$\mathbb E [x \odot \log x]$$
where $x \sim \text{Dir}(a), a\in \mathbb R^{n+1}$, $\odot$ denotes the Hadamard (or element-wise) product and the (natural) logarithm is taken component-wise.
This is an integral over the standard simplex
$$\int_{\Delta^{n}} P(x)\, x\odot \log x \:dx$$
and if we compute this component-wise (say for the last component), the problem "reduces" to
$$ C\int_0^1x_1^{a_1-1} \int_0^{1-x_1}x_2^{a_2-1} \cdots \int_0^{1-\sum^{n-1}_{i=1}x_i}x_n^{a_n-1} (1-\sum_{i=1}^n x_i)^{a_{n+1}}\log(1-\sum_{i=1}^n x_i) \:\:dx_n \cdots dx_1, C\in \mathbb R$$
Now I am really struggling to compute this integral. I have tried integration by parts on the integral over $x_n$ in two different ways, but that led nowhere.
I am wondering if there are identities or tricks that I am missing which would enable to compute this expectation?
Here is a similar question that uses identities which may help: Expected Entropy Based on Dirichlet Distribution
In fact, each component of a Dirichlet distribution follows a Beta distribution by definition
$$x_j \sim \text {Beta}(a_j, a_0-a_j)$$
where $a_0 = \sum_{i=1}^n a_i$.
Then using an identity from Wikipedia:
$$\mathbb E[x_j \log x_j]= \frac {a_j} {a_0}(\psi(a_j+1)-\psi(a_0+1)) $$
where $\psi$ is the digamma function.