Are there closed forms for the following integrals?

$$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y+y)}{y^2+\pi^2} dy, \\ I_3(w,a) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y) \cdot (a+e^y)}{y^2 + \pi^2} \, dy, \end{align}$$

where $w$ and $a$ are nonzero real parameters. I'm interested in special case of $w=1$, and $a=1$ or $a=2$.

The motivation of the problem is the Fransén–Robinson constant, and in this answer I've shown that $I_3(1,1) = \int_0^1 \frac{1}{\Gamma(x)} dx.$ But maybe there is also a closed form for $I_3(1,1)$ perhaps in term of exponential integrals.

Some related integrals.

$$\begin{align} & \int \exp\left(-e^{y}\right) dy = \operatorname{Ei}(-e^{y})+C, \\ & \int \exp\left(-we^{y}\right) dy = \operatorname{Ei}(-we^{y})+C, \\ & \int \exp\left(-e^{y}+y\right) dy = -\exp \left(-e^y \right)+C, \\ & \int \exp\left(-we^{y}+y\right) dy = \frac{-\exp \left(-we^y \right)}{w}+C, \end{align}$$

where $\operatorname{Ei}$ is the exponential integral.

  • $\begingroup$ Presumably, $w>0$ or the integral is undefined/infinite. $\endgroup$ – Thomas Andrews Sep 26 '14 at 15:45

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