I would like to compute the following tail, $$ \mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right), $$ assuming $$ \mathbb{P}[f(X_t)>x] = x^{\alpha} \log(x), $$ and $X$ is a diffusion process, meaning that $ \mathrm{dX_t}= b(X_t) \mathrm{dW_t}+ c(X_t)\mathrm{dt} $ where $W$ is a Brownian motion, and $b$ and $c$ are given functions. Thanks in advance.
