Integrating inverse functions

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I'm trying to integrate the following: $$\int_0^1 \left[\frac{c}{(1+c^{-1}(\tilde{b}))}\right]dc$$ If it helps $$c(\tilde{b})=1-\frac{1}{(1-\tilde{b})\exp\left(\frac{2\tilde{b}}{(1-\tilde{b})}\right)}$$ where $\tilde{b}\in[0,1]$ and thus $c(\tilde{b})\in[0,1]$. Any help is much appreciated