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Perhaps mathematica is not able to answer since you do not say over which ring you want your element to be integral ($\mathbb{Z} ??$). In your case, you can get conviced by typing:

sage: P = a.absolute_minpoly() ; P
x^6 - 2*x^3 + 2
sage: P(a)
0
sage: P in ZZ[x]
True
sage: P.is_monic()
True

Perhaps mathematica is not able to answer since you do not say over which ring you want your element to be integral ($\mathbb{Z} ??$). In your case, you can get conviced by typing:

sage: P = a.absolute_minpoly() ; P
x^6 - 2*x^3 + 2
sage: P(a)
0
sage: P in ZZ[x]
True
sage: P.is_monic()
True

In your case, you can get conviced by typing:

sage: P = a.absolute_minpoly() ; P
x^6 - 2*x^3 + 2
sage: P(a)
0
sage: P in ZZ[x]
True
sage: P.is_monic()
True