Ok so I tried U-subbing sqrt(x^2-4), but I didn't get anywhere. Then I though hmm, maybe trig-sub will do the trick, but I also got stuck there. I don't know what's left to do!
edit:
The integral is:
sqrt(x-sqrt(x^2-4))
what makes you think this integral has a closed-form antiderivative in the first place?
Wolfram's result ...
integrate sqrt(x - sqrt(4-x^2)) dx - Wolfram|Alpha
Wolfram DOES give me an answer...
integral[Sqrt[x - Sqrt[x^2 - 4]]] - Wolfram|Alpha
But no steps...
A shame it can't remember, but what Wolfram probably did is notice that the derivative of the integrand is virtually that integrand divided by
...
... where (key in spoiler) ...
Spoiler:
... and therefore do integration by parts (twice) on the product, ...
... (key in spoiler) ...
Spoiler:
The rest...
Spoiler:
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Don't integrate - balloontegrate!
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