Wolfram Alpha Can't Do This, Can You?

Calculus Level 5

Today, when I was in my college library, I saw two engineer students were desperate to do their calculus homework. They became more frustrated when realized even Wolfram Alpha failed to answer it. I took a look their homework in case I could solve it. This is their homework:

\[ \int_0^\infty\frac{e^{-x\sqrt{3}}}{x}\left(1-\sin x\right)\left(1+2\sin x-\cos 2x\right)\;dx=\;... \]

Could you help them out?

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