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Any hints on solving an integration of the following form,

$$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$

This arises pretty much in Poisson point processes after using the probability generating function property.

I know the above looks pretty much hopeless, I tried using Mathematica and Wolfram Alpha, both don't really help much.

I also have $x>0$ and $s>0$ real numbers if that helps.

I tried the following Wolfram|Alpha input and it gives a result in terms of $i$ imaginary, I don't understand why.

Also is there any valid approximation for which I can say that this integral can be approximately upper and lower bound by? Thanks in advance for any help.

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    $\begingroup$ i see no chance to integrate this $\endgroup$ – Dr. Sonnhard Graubner Oct 29 '14 at 19:40
  • $\begingroup$ even numerically ? @Dr.SonnhardGraubner $\endgroup$ – Tyrone Oct 29 '14 at 19:44
  • $\begingroup$ yes if $s$ is given $\endgroup$ – Dr. Sonnhard Graubner Oct 29 '14 at 19:45
  • $\begingroup$ You were given the first part in a previous question in terms of the Ei function, if I remember correctly, so why include it again? $\endgroup$ – Did Oct 29 '14 at 19:49
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    $\begingroup$ A nicer representation of the integral can be found by making the obvious substitution $t = \sqrt[4]{y}$ then $I = 4s\int_{\sqrt[4]{x}}^\infty \frac{1}{s+t^4}\left(t^3e^{-t^2} + t - te^{-t}\right)dt$. From this it seems hopeless to get an analytical answer I'm afraid. Perhaps when $x=0$ and some special values for $s$. Try numerical integration! $\endgroup$ – Winther Oct 29 '14 at 19:56

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