$$\mathfrak{E} = \int \exp \Big[ \operatorname{erf}{\Big(\frac{x}{u} \Big)} \Big] \, dx $$
This integral has come up in my research and I have not found any known solutions (integral tables, wolfram alpha, matlab symbolic integration). I am still intrigued as the function has interesting characteristics that seem like it would be common in other physical systems. Plus, $\mathfrak{E}$ is continuously differentiable, which seems promising. But my math capabilities have not sufficed in pursuit of a solution. Has anyone out there dealt with this integral before or seen it come up?
Any answers or comments are much appreciated!
EDIT: Many thanks to Robert for pointing out my misuse of the theorem of Liouville (removed) and for Michael's help with my formatting!