How to evaluate the integral $\int_{-1}^{0}\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{\cdots\sqrt{1+x}}}}}dx=?$

Question

We have :

$$\int_{-1}^{0}\sqrt{1+\sqrt{1+\sqrt{1+x}}}dx=\frac{8}{315}\sqrt{2}\Big(16+\sqrt{233+317\sqrt{2}}\Big)$$

We are lucky because this integral have an anti-derivative like here.

More seriously I have tried the following substitution : $x=((t-1)^2-1)^2-1)^2-1$

We get a big polynomial easily integrable .

But what about the following integral : $$\int_{-1}^{0}\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{\cdots\sqrt{1+x}}}}}dx=$$

Have you other technics to evaluate this ?

Thanks in advance for your contributions !

Update :

As it seems to be unclear I ask for a finite number of nested radical.Furthermore it's not an innocent question because I know that it converges to the golden ratio .The imlpicit question is : How to get nested radicals near from the value of the golden ratio ?

Thanks again.