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I am looking for a solution or a method of approximation for : $$\int \frac{1}{1-we^w}dw$$ that came up while working on an ODE problem.

Got any suggestions?

Note: $w$ is also a one variable function

Thanks to anyone who can lend a hand

Update:

The original ODE is: $$xdw=(e^{-w}-w)dx$$

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    There is no solution in elementary functions. The methods of approximations will depend on the restrictions on the variable (for example if $|w|<1$ a series solution seems best) – 2017-01-29
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    What's the original ODE? Perhaps you made an error? – 2017-01-29
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    Updated the question with original ODE, thanks to both for taking a moment of your time to help me out – 2017-01-29
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    Kind of important, there is a constant solution, there is a real number $w_0$ with $e^{-w_0} = w_0.$ Solutions for first order ODE cannot cross, so there are solutions with $w > w_0$ and others with $w < w_0.$ You should expect some horizontal asymptotes at $w \rightarrow w_0,$ along with evident vertical asymptotes at $x \rightarrow 0$ – 2017-01-29

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