Question

I want to integrate 〖e^(e^(ix)-ix) or
exp(exp ix -ix)

Answer 1

i think no closed form

Answer 2

Letting u=e^ix you get \(-i\int \frac{e^u}{u^2}\) and then integrate by parts you get \(-i\frac{e^u}{u^2}-i\int\frac{e^u}{u}\). The second term is the exponential integral \(\text{Ei}(u)\).
http://en.wikipedia.org/wiki/Exponential_integral

Answer 3

Are you integrating over a certain contour? For some paths there are no closed forms (in terms of elementary functions), but for others there are, especially in limiting cases..