expectation of lgamma of gamma distribution

Question

Is there a closed-form expression for $E[\log(\Gamma(X))]$, where $X \sim Gamma(k, \theta)$?

Edit: Note the gamma function inside the log.

Edit 2: If there's no closed-form expression, is there a better way to approximate this expectation than monte carlo simulation of X?

Answer 1

The delta method gives $$\operatorname{E}[f(X)] \approx f(\operatorname{E}[X]) + \frac{f''(\operatorname{E}[X])}{2} \operatorname{Var}[X]$$ so for $f(x) = \log \Gamma(x)$ it is not difficult to obtain the approximation $$\operatorname{E}[\log \Gamma(X)] \approx \log \Gamma(k \theta) + \frac{1}{2} k\theta^2 \psi^{(1)} (k\theta),$$ where $\psi^{(1)}$ is the second derivative of the logarithm of the gamma function.

Answer 2

The answer is written out here (https://www.physicsforums.com/threads/mean-and-variance-of-loggamma-distribution.779681/), except the final arithmetic.

What seems like related parameterization from Mathematica: https://reference.wolfram.com/language/ref/LogGammaDistribution.html