Consider the following integral :
$$I=\int_0^\infty \frac {e^{c_1u}e^{-c_2e^{cu}}}{1-bu^2}du$$
Where , $c_1, c_2, c, b $ are positive constant
Expanding the denominator term into geometric series and integrating term by term doesn't help the purpose w.r.t computation.
I'm really hoping for closed form (or at least partial fraction decomposition (obviously in terms of b) - finite or infinite - Mittag - Leffler)
Question : Is there a closed form representation for $I(c_1, c_2, c, b )$? And if yes , how to evaluate it?
How to convert this integral into complex integral?