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I was trying to solve the integration of $\int \sin(x) e^{\sin(x)-x} (\sin(x)-x) dx$ with integration by parts, but with no success.

I also tried to expend the expression to:

$\int \sin^2(x) e^{\sin(x)-x} dx - \int x\sin(x) e^{\sin(x)-x} dx$

Also this didn't really help. Maybe someone has better ideas?

robjohn
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ChunleiA
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  • where did you found this integral? – Dr. Sonnhard Graubner Apr 10 '15 at 11:56
  • It doesn't have any antiderivative. Even W|A can't integrate it. – Prasun Biswas Apr 10 '15 at 11:59
  • @PrasunBiswas I have seen many times in the past where a solution exists that wolfram cannot find... – bjd2385 Apr 10 '15 at 12:10
  • @ Dr. Sonnhard Graubner， the integral is the outcome of a mathematical model, which intends to describe the average number of neighbors of a single node/host in a given network. – ChunleiA Apr 10 '15 at 12:26
  • @jm324354, totally agree. if W|A could solve this directly, then I don't have to post it here ;) Also tools like wxMaxima could not solve it directly. But I think with some steps of simplifications it should be solvable by the machines ... – ChunleiA Apr 10 '15 at 12:31
  • @ChunleiA Have you thought about applying numerical techniques? – Matthew Cassell Apr 10 '15 at 12:34
  • I also tried to learn something from one of the previous posts here: http://math.stackexchange.com/questions/636803/how-to-integrate-a-three-products, which seems a bit similar to my problem. So far it doesn't work out. – ChunleiA Apr 10 '15 at 12:34
  • @Mattos, thanks for the hint. I don't have knowledge about numerical techniques, but now I am looking for some materials about it. – ChunleiA Apr 10 '15 at 12:44
  • The only way in which this integral might possess a closed form is if we were to add $0$ and $\infty$ as limits of integration, and then try to express it in terms of Bessel and Struve functions. – Lucian Apr 10 '15 at 16:47

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