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?
