\[\Large\int_0^1 \exp({x+e^x+e^{e^x}+e^{e^{e^x}}}) \, dx\]

If the above integral can be expressed as

\[^Ae - ^B e\]

for positive integers \(A\) and \(B\), find \(A+B\).

**Notations**:

\(\exp(x) \) denotes the exponential function, \(\exp(x) = e^x \).

\(^na\) denotes the Tetration function, \(\Large ^n a = \underbrace{a^{a^{a^{\cdot^{\cdot^a}}}}}_{n\text{ number of } a\text{'s}} \).

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