# Does this random variable built from log normals have a recognizable distribution?

Let $X_t$ be a standard Brownian motion, and $\lambda$ and $\alpha$ are positive constants. Consider the random variable: $$\int_0^t \lambda e^{\lambda u + \frac{\alpha u X_t}{t} - \frac{\alpha^2 u^2}{2t}} du$$

Does this guy have a standard distribution? Thanks!

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