I'm looking for a solution to the following integral.


$$\int\_{\lambda}^{y}(x-a)^{-b}x^{-c}\exp\left( -d x^{-e} \right)dx,$$
where $b,c,d,e> 0$ and $0< a < \lambda < y$.


This equation appears in the context of Physical Layer Security, which is an area of study in digital communications (telecom).



Is there a solution to $\int_{\lambda}^{y}(x-a)^{-b}x^{-c}\exp\left( -d x^{-e} \right)dx$?
There is no closed-form expression in terms of known functions, but if $a$ is small, you could use the power series in terms of the [exponential integral](https://en.wikipedia.org/wiki/Exponential_integral#Generalization),
$$\int\_{\lambda}^{y}(x-a)^{-b}x^{-c}\exp\left( -d x^{-e} \right)\,dx=\sum\_{p=0}^\infty \frac{b\Gamma(p+b)}{p!\Gamma(1+b)}\frac{a^p}{e}
\left[y^{-k}E\_{1-k/e}\left(d y^{-e}\right)-\lambda^{-k}E\_{1-k/e}\left(d \lambda^{-e}\right)\right].
$$
where $k=b+c+p-1$.


