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I seem to have seen quite a lot of integrals in the form:

$$\int_0^\infty \frac{\text{d}x}{e^x+(1+x^n)}$$

But none of those hold a closed forms (at least to my knowledge)

$$\Large\color\red{\int_0^\infty \frac{\text{d}x}{e^x+x^n}}$$

Does a closed form exist of this integral? (in terms of $n$)

  • we don't even have a closed form formula for the poles and residues of $1/(e^z + z^n)$ for $n \ne 0$ – reuns Jan 14 '16 at 00:46
  • @user1952009 u can give the poles in terms of Lambert W function. – tired Jan 14 '16 at 01:18
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    I seem to have seen quite a lot of integrals in this form - Where ? – Lucian Jan 14 '16 at 08:52
  • @Lucian I was investigating different ways to evaluate $\int_0^\infty \frac{, dx}{e^x(x^n+1)}$ and this keeps coming in the way. –  Jan 14 '16 at 15:44
  • @JohnVine: That expression does not possess an elementary closed form even for “nice” values of $n\neq0.$ – Lucian Jan 14 '16 at 17:03
  • I know, I was wondering if the modification I made (not distributing) would make a difference. I miswprded my question @Lucian . Am editing –  Jan 14 '16 at 17:50

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