Why substitution method does not work for $\int (x-\frac{1}{2x} )^2\, \mathrm dx$?

Question

Why $$\int \ \left(x-\frac{1}{2x} \right)^2 \, \mathrm dx$$ is easy to integrate once $$\left(x-\frac{1}{2x} \right)^2$$ is expanded, but impossible using substitution method? (tried 5 different subs but of course that is not the proof that there is no suitable substitution) if mathematical results are independent of the logical methods used to derive them, why something so simple works one way but not the other?

Answer 1

The situation you describe is not too dissimilar to the following: imagine standing in front of a wall with a few doors in it. You want to go in. At first you don't notice any doors so you wander up and down in front of the wall, until eventually you realize there was a door right in front of you all along. Then since you believe there's only one room inside, you are surprised that there wasn't a door everywhere else on the wall outside. Does this make clearer why your surprise is unfounded?

Put another way, equivalence of any correct route doesn't show existence of any proposed route.