Let $B_t$ be a Brownian motion with variance 1. We know that $\int_0^1 B(t) \mathrm{d} t \sim \mathcal{N}(0,1/3)$. I am interested to know what we can say about the law of the two random variables $X = \int_{0}^1 B(t)^2 \mathrm{d}t = \langle B,B\rangle_{L^2([0,1])}$ and $Y = \int_{0}^{1} \left( B(t) - \int_0^1 B(s)\mathrm{d}s \right)^2 \mathrm{d}t = X - \left( \int_0^1 B(s)\mathrm{d}s \right)^2$.

Is there a well-known probability law hidden behind them?

Thank you for attention.