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I am given the integral
$$ \int_{-\infty}^{\infty} \frac{1}{\left(e^t - t\right)^2 + \pi^2} \, \mathrm{d}t $$
and have good reason to believe there is some sort of closed-form solution. However, all standard (and some non-standard) techniques of evaluation seem to come up short. Does anyone know how to evaluate it or have a reference for it?

Remark. This is my first post on MSE! Please forgive me if I've made any mistakes regarding etiquette or have committed some other impropriety.

Ian
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    Welcome to MSE! Can you add a bit about why you have good reason to believe a closed-form solution exists? It would also be worth mentioning which techniques of evaluation you've tried already, and why they've come up short. – Carl Schildkraut Jun 08 '25 at 05:38
  • @AmrutAyan Thanks for the link to that post. That resolves my question well enough. Also, thanks for the response, Carl Schildkraut! Some of the techniques I tried were contour integration (which I am admittedly not great at) and converting the inside into a new integral with the inverse Laplace transform of $\frac{1}{s^2+a^2}$ and swapping the integrals. P.S. I think I recognize your name from a bunch of (excellent) problems I did a long time ago. – Ian Jun 08 '25 at 05:49

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