Whenever there are multiple radical terms in the denominator, the process of simplification becomes quite a challenge, but I realized after some time that the denominator 1−3√3+3√9=(3√3)2−3√3+1 looks to take the form of the beginnings of the factorization of the sum of cubes.
In the general factorization, a3+b3=(a+b)(a2−ab+b2). If we consider the case where a=3√3 and b=1, then we can multiply the denominator by 3√3+1 to transform the denominator into a sum of cubes and then simplify.
41−3√3+3√9∗3√3+13√3+1=4(3√3+1)(3√3)3+13=4(3√3+1)3+1=3√3+1