Name of a distribution similar to the exponential for a simulation I'm using the continuous distribution
$$F(x)=1-(1+x)e^{-cx} $$ for $x\geq 0$ with $c\geq 1$.
Do you know if this distribution has a name?

The corresponding density function can be found to be
$$ f(x)= (cx+ c-1) e^{-cx}, \quad x\ge 0 $$
which can be written as a mixture
$$ cx  e^{-cx} + (c-1)  e^{-cx} $$ which is a mixture of two gamma distributions, assuming $c\ge 1$ (necessary to avoid negative mixture weights). But from the form of the cdf (cumulative distribution function) given in the post, we can see that $c>0$ is enough, so that, interestingly, the mixture representation is not possible for all $c$.
There is probably no specific name for this distribution, at least, I did some searching and could not find any.
