How to find PDF and expected value of $\max(x,0)$ for random variable $x$

Let $$a,b,c, \tau$$ be positive constants and $$x$$ is an exponentially distributed variable with parameter $$\lambda = 1$$, i.e. $$x\sim\exp(1)$$.

$$E = \tau\Big[a\frac{1+a}{1+e^{-bx+c}} - 1 \Big]^+$$

where $$[z]^+ = \max(z,0)$$

How can I find

1. The PDF for $$E$$
2. The expectation of $$E$$.

first calculate the interval where $$Z=\frac{a(a+1)}{1+e^{-bX+c}}-1\leq0$$
The probability of this interval is a mass probability assigned to $$Z=0$$. The remainig probability is distributed over $$Z$$ according to the specified random law. The resulting "PDF" is a mixed density, not absolutely continuous.
Finally calculate the density of $$Y=\tau[Z]^+$$ by a monotonic transformation and its expectation in the way you are used to do