Decompose Polynomials Given an integral polynomial of degree strictly greater than one, completely decompose it into a composition of integral polynomials of degree strictly greater than one.
Details

An integral polynomial is a polynomial with only integers as coefficients.
Given two polynomials p and q the composition is defined by (p∘q)(x):=p(q(x)).
The decomposition of an integral polynomial p is a finite ordered sequence of integral polynomials q1,q2,...,qn where deg qi &gt; 1 for all 1 ≤ i ≤ n and p(x) = q1(q2(...qn(x)...)), and all qi are not further decomposable. The decomposition is not necessarily unique.
You can use e.g. lists of coefficients or built in polynomial types as input and output.
Note that many builtins for this task actually decompose the polynomials over a given field and not necessarily integers, while this challenge requires a decomposition integer polynomials. (Some integer polynomials might  admit decomposition into integer polynomials as well as decomposition that contain rational polynomials.)

Examples
x^2 + 1
[x^2 + 1] (all polynomials of degree 2 or less are not decomposable)
x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6 x - 1
[x^3 - 2, x^2 - 2x + 1]
x^4 - 8x^3 + 18x^2 - 8x + 2 
[x^2 + 1, x^2 - 4x + 1]
x^6 + x^2 + 1
[x^3 + x + 1, x^2]
x^6
[x^2, x^3]
x^8 + 4x^6 + 6x^4 + 4x^2 + 4 = (x^2 + 1)^4 + 3
[x^2 + 3, x^2, x^2 + 1]
x^6 + 6x^4 + x^3 + 9x^2 + 3x - 5
[x^2 + x - 5, x^3 + 3*x], [x^2 + 5*x + 1, x^3 + 3*x - 2]

Use Maxima for generating examples: Try it online!
Some decomposition algorithms can be found here and here.
Pari/GP, 84 bytes
f(p)=[if(q'',[f(q),r],p)|r&lt;-x*divisors(p\x),r''&amp;&amp;p==subst(q=substpol(p,r,x),x,r)][1]

Based on the algorithm described here.
Try it online!