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Formally, $$(x^4-x)^{1/3} = -(x-x^4)^{1/3}= -x^{1/3}\; (1-x^3)^{1/3}\ ,$$ so the problem is the last factor.
(We may substtitute $y=x^{1/3}$ and calculate w.r.t. $y$ if the other factor is also a problem.)
Then the code
PREC = 20 # feel free to take 100 or more
R.<x> = PowerSeriesRing( QQ, default_prec=PREC )
# (1-x^3)^(1/3) does not work, but let us try a version of exp log (1-x^3)^(1/3)
f = exp( 1/3 * log( 1-x^3 ) )
print f
finds
1 - 1/3*x^3 - 1/9*x^6 - 5/81*x^9 - 10/243*x^12 - 22/729*x^15 - 154/6561*x^18 + O(x^20)
and we can check it rapidly:
sage: f^3
1 - x^3 + O(x^20)
The power series inverse with respect to the multiplication is:
sage: 1/f
1 + 1/3*x^3 + 2/9*x^6 + 14/81*x^9 + 35/243*x^12 + 91/729*x^15 + 728/6561*x^18 + O(x^20)
Or directly:
sage: exp( -1/3 * log( 1-x^3 ) )
1 + 1/3*x^3 + 2/9*x^6 + 14/81*x^9 + 35/243*x^12 + 91/729*x^15 + 728/6561*x^18 + O(x^20)
