The trig functions sin(t) and cos(t) = sin'(t) parametrize the curve x^2 + y^2 = 1. There is a nonnegative integer associated to curves defined by a polynomial equation, called its genus, and the reason you can successfully integrate certain expressions involving square roots is because of the trick of completing the square and the fact that the curve x^2 + y^2 = 1 of degree 2 has "genus 0". This means that curve can be parametrized by rational functions -- look up the Weierstrass or tan(t/2) substitution -- and what's happening with trig substitution is that you're sort of integrating rational functions in disguise.

In higher degree this whole chain of ideas breaks down. First of all, there is no such general trick as "completing the cube" (cubics have too many parts) and for n > 2 the curve x^n + y^n = 1 has positive genus, which means it can *not* be parametrized by rational functions. To use your terminology, there is no "easy" antiderivative of (1-x^(3))^(1/3). That is a theorem, not a reflection of some lack of creativity on mathematician's part.

There are analytic functions describing the points on genus 1 curves, and these functions are called *elliptic* functions (the name "elliptic" is used for historical reasons, and these functions don't parametrize an ellipse). They are a sophisticated generalization of trigonometric functions. But it is not anything that anyone taking calculus is going to find user friendly at all.