E[exp(-X)] when X ~ Weibull

Question

Let $f(x) = ckx^{k-1}\exp(-cx^k)$. Can we then derive an analytical expression for the following integration?

$\int_0^z f(x) \exp(-x) dx = \int_0^z ckx^{k-1}\exp(-cx^k) \exp(-x) dx$

Answer 1

I think that the presence of $e^{-(x+cx^k)}$ makes the problem difficult (not to say impossible). The only cases I was able to solve corresponds to cases where we can complete the square (using eventually changes of variable).

If $$I_k= \int_0^z c\,k\,x^{k-1}\,e^{-(x+cx^k)}\, dx$$ the only cases I was able to solve are $$I_1=\frac{c \left(1-e^{-(c+1) z}\right)}{c+1}$$ $$I_2=1-e^{-z (c z+1)}+\frac{\sqrt{\pi } e^{\frac{1}{4 c}} \left(\text{erf}\left(\frac{1}{2 \sqrt{c}}\right)-\text{erf}\left(\frac{2 c z+1}{2 \sqrt{c}}\right)\right)}{2 \sqrt{c}}$$ $$I_{1/2}=\frac{1}{2} \sqrt{\pi } c e^{\frac{c^2}{4}} \left(\text{erf}\left(\frac{c}{2}+\sqrt{z}\right)-\text{erf}\left(\frac{c}{2}\right)\right)$$

If we consider $$J_k= \int_0^\infty c\,k\,x^{k-1}\,e^{-(x+cx^k)}\, dx$$ the result write as a series of generalized hypergeometric and Airy functions (and the results are not looking very appealing at all !). The simplest $$J_3=\, _1F_2\left(1;\frac{1}{3},\frac{2}{3};-\frac{1}{27 c}\right)-\frac{2 \pi }{3 \sqrt[3]{3c} }\text{Bi}\left(-\frac{1}{\sqrt[3]{3c} }\right)$$