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Recently puzzling around with the Lambert-$W$ function, I have become intrigued by the integrals:

$$\int e^{xe^{x}}dx \tag{1}$$

$$\int e^{xb^{x}}dx \tag{2}$$

$$\int e^{axb^{x}}dx \tag{3}$$

the latter two with $0 < b < 1$.

Are any of these well-known indefinite integrals with closed-form solutions?

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    $\begingroup$ No as far as I know. But proving my claim is a hard task. You can get an idea about such a proof from here. $\endgroup$
    – Bumblebee
    Jun 27, 2020 at 6:41

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