At a first glance, the golden ratio lurking there may seem indeed a curious phenomenon. But everything becomes OK looks more regular observing that the RHS of the equality is a partial fraction decomposition, and that the golden ratio is the a root of very simple polynomials, not unlikely to appear as denominators (here the derivative fraction is $\frac{x^2-1}{x^2+(x^2+1)^2}$ &c).
At a first glance, the golden ratio lurking there may seem indeed a curious phenomenon. But everything becomes OK observing that the RHS of the equality is a partial fraction decomposition, and that the golden ratio is the root of very simple polynomials, not unlikely to appear (here the derivative is $\frac{x^2-1}{x^2+(x^2+1)^2}$ &c).