# How to solve $\int \sin(x) e^{\sin(x)-x} (\sin(x)-x) dx$

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?

• 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... – jm324354 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