Q:

Help with a very hard integral

$$
\int^{\infty}_{1} \Bigl(\,\Bigl|3\mathrm{e}^{-\tfrac{x^2}{\sqrt{x-1}}}+x\Bigr|-\Bigl|3\mathrm{e}^{-\tfrac{x^2}{\sqrt{x-1}}}-x\Bigr|\,\Bigr)\,\mathrm{d}x=
\int^{\infty}_{1} \Bigl(3\mathrm{e}^{-\tfrac{x^2}{\sqrt{x-1}}}+x-\Bigl|3\mathrm{e}^{-\tfrac{x^2}{\sqrt{x-1}}}-x\Bigr|\,\Bigr)\,\mathrm{d}x$$
I literally have no idea on how to do this, perhaps through complex analysis or polar coordinates(?).  Also I only want an exact answer.  
Thanks! 
Edit:
Thanks to metamorphy for pointing out that I don't need the first absolute value. 

A:

The first step is proving $3\exp -\frac{x^2}{\sqrt{x-1}}<x$ for all $x\ge 1$. (Judging by this graph, you just need to check its derivative is negative.)  So your integral is $$\int_1^\infty \left(3\exp -\frac{x^2}{\sqrt{x-1}}+x+3\exp -\frac{x^2}{\sqrt{x-1}}-x\right)dx=6\int_1^\infty\exp -\frac{x^2}{\sqrt{x-1}} dx.$$However, there doesn't seem to be a non-numerical method for this.