We study a Brownian motion with drift in a wedge of angle $β$ which is obliquely reflected on each edge along angles $\varepsilon$ and $δ$. We assume that the classical parameter $α=\frac{δ+\varepsilon - π}β$ is greater than $1$ and we focus on transient cases where the process can either be absorbed at the vertex or escape to infinity. We show that $α\in\mathbb{N}^*$ is a necessary and sufficient condition for the absorption probability, seen as a function of the starting point, to be written as a finite sum of terms of exponential product form. In such cases, we give expressions for the absorption probability and its Laplace transform. When $α\in\mathbb{Z}+\fracπβ\mathbb{Z}$ we find explicit D-algebraic expression for the Laplace transform. Our results rely on Tutte's invariant method and on a recursive compensation approach.