Second derivative of an integral at zero

Let $f:[0,1]\to\mathbb R$ be a function defined as

$$f(t)=\begin{cases}t^3\left(1+\frac15\cos(\log_et^4)\right), & t\in(0,1],\ 0, & t=0.\end{cases}$$

Let $F:[0,1]\to\mathbb R$ be defined as $F(x)=\int_0^xf(t)\,dt$. Then $F"(0)$ equals