This is exactly the so-called "Renyi parking process." Renyi proved
$$\varphi(x) = cx + c - 1 + o(1)$$
where $$c = \int_0^\infty \exp\left( -2 \int_0^x \frac{1 - e^{-y}}{y}\,dy \right)\,dx \approx 0.7475979203\,.$$
The $o(1)$ term is in fact $O( (2e/x)^{x - 3/2})$. See this recent reference by Clay and Simanyi. Lines (1.3) and (1.4) show the form above, and line (1.1) is exactly the integral expression you derive.