calculus and analysis - Integrate over a piecewise function

I want to calculate the indefinite Integral of

$$f(x)=\begin{cases} 2x\cos(\frac{1}{x})& \text{ if } x\ne 0 \\ 0& \text{ if } x=0 \end{cases}.$$

I use the following code:

F[x_] := Piecewise[{0, x==0}, {2*x*Cos[1/x], x != 0}];
Integrate[Piecewise[{{0, x == 0}, {2*x*Cos[1/x], x != 0}}], x]

It doesn't evaluate. I don't know why it doesn't evaluate the piecewise function.

Maybe this is quite simple question, but it's not easy for me, a greenest amateur.