Non-Elementary Integration · Challenge problem

Let $F(a)=\displaystyle\int_0^{\infty}e^{-x}\,\dfrac{1-\cos(ax)}{x}\,dx$. By differentiating under the integral sign (Feynman's trick), evaluate $F(3)=\displaystyle\int_0^{\infty}e^{-x}\,\dfrac{1-\cos 3x}{x}\,dx$.