If $\mathbf{x}\sim\mathbf{N}(\mathbf{0},\mathbf{I})$, and assume that $A$ is a symmetric positive definite matrix, how can I calculate the following two expectations where there is a logarithm in it? $$ \mathrm{E}_\mathbf{x}\left[\log(\mathbf{x}^\top A\mathbf{x})\right] $$ and

$$ \mathrm{E}_\mathbf{x}\left[(\mathbf{x}^\top A\mathbf{x})\log(\mathbf{x}^\top A\mathbf{x})\right] $$

Using the decomposition $A=U\Lambda U^\top$ and $\mathbf{y}=U^\top\mathbf{x}$, the first expectation can be reduced to $$ \mathrm{E}_\mathbf{y}\left[\log(\mathbf{y}^\top \Lambda\mathbf{y})\right] $$

$$ =\mathrm{E}_\mathbf{y}\left[\log\left(\sum_i{\lambda_i\mathbf{y}_i^2}\right)\right] $$ where $y$ has the same distribution as $x$ because $U$ is a orthogonal matrix.

However, since the random variables $y_i$ are in the logarithm function, I cannot decompose the expectation further and such a expectation seems to be difficult to calculate.

Could you please help me with this problem or give some suggestions about it? Thank you very much!