4f2de1…: “Over the quiet quadrilateral”

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Let $S= \{(x,y,z):-1\le x\le 1,-1\le y\le 1, -1 \le z\le 1\}$ . Let $F : \R ^3 \to \R$ be the function defined by the rule $F(x,y,z) = xyz + x^3 y^3 z^3 + x^5 y^5 z^5$ Evaluate $\iiint _S F \d V$ without doing too much work, noting that $F(x,y,z) = - F(-x,y,z)$ .

$\iiint _S F \d V = \answer {0}.$