## Inverse function of $y=W(e^{ax+b})-W(e^{cx+d})+zx$

I would like to find the inverse of the following function

$$y=W(e^{ax+b})-W(e^{cx+d})+zx$$

where $W$ is the Lambert $W$ function and $a,b,c,d,z$ are some constants.

I asked the same question here:

http://math.stackexchange.com/questions/201634/inverse-function-of-y-weaxb-wecxdzx

I also checked some papers but couldnt decide how to proceed, though I have the following ideas:

$1$- Is it possible to write $$W(e^{f(x)})=W(e^{ax+b})-W(e^{cx+d})$$

$2$- Having $$y_1=W(e^{ax+b})+z_1x$$ and $$y_2=-W(e^{cx+d})+z_2x$$

where $z=z_1+z_2$, $y=y_1+y_2$ and $f^{-1}(y_1)$ and $f^{-1}(y_2)$ are known functions as already found. Can we say that $f^{-1}(y)=f^{-1}(y_1)+f^{-1}(y_2)$? or can we modify this idea to get something useful?

Any help will be appreciated. Thanks in advance.

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