Is there any integral for the Golden Ratio?

I was wondering about important/famous mathematical constants, like $$ee$$, $$π\pi$$, $$γ\gamma$$, and obviously the golden ratio $$ϕ\phi$$.
The first three ones are really well known, and there are lots of integrals and series whose results are simply those constants. For example:

$$π=2e+∞∫0cos(x)x2+1 dx \pi = 2 e \int\limits_0^{+\infty} \frac{\cos(x)}{x^2+1}\ \text{d}x$$

$$e=+∞∑k=01k! e = \sum_{k = 0}^{+\infty} \frac{1}{k!}$$

$$γ=−+∞∫−∞x ex−ex dx \gamma = -\int\limits_{-\infty}^{+\infty} x\ e^{x - e^{x}}\ \text{d}x$$

Is there an interesting integral* (or some series) whose result is simply $$ϕ\phi$$?

* Interesting integral means that things like

$$+∞∫0e−xϕ dx\int\limits_0^{+\infty} e^{-\frac{x}{\phi}}\ \text{d}x$$

are not a good answer to my question.

Potentially interesting:

Perhaps also worthy of consideration:

A development of the first integral:

which stem from the relationship $(x-\varphi^m)(x-\bar\varphi^m)=x^2-(F_{m-1}+F_{m+1})x+(-1)^m$, where $\bar\varphi=\frac{-1}{\varphi}=1-\varphi$ and $F_k$ is the $k$th Fibonacci number. I particularly enjoy: