How to evalutate $ \int_{\Bbb V}{\arctan(x y^2 z^3)} \, dx \, dy \, dz $ where $ \Bbb V = \{ x^2 + y^2 \le 1, \, 0 \le z \le x^2 \} $?

Question

Consider: $ \Bbb V = \{ x^2 + y^2 \le 1, \, 0 \le z \le x^2 \} $, then evalutate the integral: $$ \int_{\Bbb V}{\arctan(x y^2 z^3)} \, dx \, dy \, dz $$

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My attempt:

1. The function is such that $f(x,y,z)=-f(-x, y,z)$;
2. The domain is such that $(x,y,z) \in \Bbb V \Rightarrow (-x,y,z) \in \Bbb V$;
   -since $(x)$ appears only squared: even the interval of $(z)$ doesn't distinguish $(x)$ from $(-x)$

Thus, the integral is null by symmetry.

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I tried to compute this integral with Wolfram Cloud to check the result, by this code:
\[ScriptCapitalR] = ImplicitRegion[x^2 + y^2 <= 1 && 0 <= z <= x^2, {x, y, z}]; Integrate[ArcTan[x y^2 z^3], {x, y, z} \[Element] \[ScriptCapitalR]]

But I got: "This computation has exceeded the time limit for your plan"

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Can anyone confirm (or refute) my attempt?