MHB Calculate Integral: $\int_0^1 \frac{(1-x)e^x}{x+e^x}dx$

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Calculate the following integral
$$\int\limits_0^1\frac{(1-x)e^x}{x+e^x}dx$$
 
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From what problem did this integral arise? It doesn't seem to succumb to any standard techniques in regular calculus. You might be able to try contour integration, or maybe differentiation under the integral sign. Or maybe even Laplace Transforms.
 

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