Is there any integral for the Golden Ratio?

I was wondering about important/famous mathematical constants, like e, π, γ, and obviously the golden ratio ϕ.
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

e=+k=01k!

γ=+x exex dx

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

* Interesting integral means that things like

+0exϕ dx

are not a good answer to my question.

Answer

Potentially interesting:

logφ=1/20dxx2+1

Perhaps also worthy of consideration:

arctan1φ=2011+x2dx20dx=2211+x2dx22dx

A development of the first integral:

logφ=12n1F2n+F2n220dxx2+1

logφ=12nF2n+1+F2n121dxx21

which stem from the relationship (xφm)(xˉφm)=x2(Fm1+Fm+1)x+(1)m, where ˉφ=1φ=1φ and Fk is the kth Fibonacci number. I particularly enjoy:

logφ=1320dxx2+1
logφ=1691dxx21
logφ=19380dxx2+1
logφ=1121611dxx21

Attribution
Source : Link , Question Author : Elliptic Curve , Answer Author : Community

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