This seems too straightforward for perplexus. I hope I missed some clever trick.

Just square each side and make a 6th power polynomial:

x^6-6x^4+9x^2-x-2 = 0

Factoring yields:

(x-2) * (x^2+x-1) * (x^3+x^2-2x-1) = 0

A little help from a numeric solver gets us roots of:

2, (-1-sqrt(5))/2, (-1+sqrt(5))/2, -1.8019, -0.44504, 1.2470

Plugging them back into the original gets three of the six as final answers:

x = {2, (-1+sqrt(5))/2, -0.44504 }