Find the marginal PDF of $\theta_1$ such that $\theta_1 \le \theta_2 \le \theta_3$ where $ \theta_i\sim Beta(\alpha_i,\beta_i)$
$f_{\theta_1} = \int_{{\theta_3}=\theta_1}^1 \int_{{\theta_2}=\theta_1}^{\theta_3} f(\theta_1, \theta_2, \theta_3)\ \times \frac{1}{P(\theta_1 \le \theta_2 \le \theta_3)} d{\theta_2}d{\theta_3} $
$f(\theta_1, \theta_2, \theta_3)$ is the joint beta distribution.
Please see following link: Integral of Joint Beta Distribution
Can I find the maximum of this marginal pdf without solving for the integral?
If there is no bounded solution, can we use simulation approach?