Solving special exponential integral

[simone8888]

Has anybody got an idea how to solve this integral. I tried integration by parts, and that made the things even more complicated, substitutions as well. I used Mathematica to Solve that problem. Here is the integral:

https://scontent-b-ams.xx.fbcdn.net/hphotos-ash3/t1/1010082_10202689859479307_236334784_n.jpg


Or in Mathematica, it would be:


Integrate[E^(-((x^2 b c^2)/(x^2 + b)))/(x^2 + b)^2,x]

 

[D H]

http://www.wolframalpha.com/input/?i=Integrate[E^(-((x^2+b+c^2)/(x^2+++b)))/(x^2+++b)^2,x]
When Mathematica says "no result found in terms of standard mathematical functions" the odds are pretty good that the function to be integrated doesn't have a nice closed form solution.

Even relatively simple integrals such as $\int \exp(-x^2)\,dx$ and ##\int \exp(x)/x\,dx don't have closed form solutions in the elementary functions. However, these two integrals raise their heads so often that special functions have been invented to represent these integrals. Just by appearance, yours almost certainly is not integrable in the elementary functions, and because it's rather esoteric, I doubt there is an existing special function that can represent it, either.

There are lots of numerical techniques to solve such integrals when expressed as a definite integral. The generic name for such numeric integration is quadrature.