Since $f$ is deterministic, we get that $X_{t}$ is Gaussian https://math.stackexchange.com/questions/3221012/let-x-t-int-0t-sigma-tdb-t-what-is-the-law-of-x-t?noredirect=1&lq=1

with variance  $\mathbb{E}\int_0^t f(s)^2 ds$.

This process is just a time changed Brownian motion see here https://almostsuremath.com/2010/04/20/time-changed-brownian-motion/