Derivative nonlinear Schrodinger equation

Question

I'm dealing with the following DNLS $$iu_t+u_{xx}=i(|u|^2u)_x$$ Let's consider the following transformation $w=\exp(-i\int_{-\infty}^x|u|^2dy)u$. I'm interested in the equation satisfied by $w$. I should get $$iw_t+w_{xx}=-iw^2\partial_x\bar{w}-\frac{1}{2}|w|^4w$$ But making the computations I get another expression. I've found the following: $$w_t=\exp(-i\int_{-\infty}^x|u|^2dy)u_t-iu\int_{-\infty}^x\partial_t(|u|^2)dy\exp(-i\int_{-\infty}^x|u|^2dy)$$ and analogous expressions for the other quantities appearing in the DNLS; putting all togheter I don't get the desired formula.I wrote only the expression for $w_t$ because I think it contains the problem. Does it contain some mistakes? Any suggestions will be apprecieted.