How to integrate $\int e^{xe^{x}} \mathrm{d}x$?

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Is it possible to integrate the following function analytically? I've been trying integration by parts, but it doesn't work.

$$ \int e^{xe^{x}} \mathrm{d}x $$

Thanks in advanced.

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There is no elementary antiderivative here. That can be proved using the Risch algorithm.