Yes. You do know about the 'show steps' button on Wolfram, I suppose? But it doesn't seem to be coming up with the goods, that's true: integrate 1/((x^2 + a^2)*Sqrt[x^2 + a^2 + b^2]) - Wolfram|Alpha.

Well, you can always differentiate the integral! The inside function of the main chain rule process is clearly enough,

... so we could fantasise about having had the inspiration to manipulate the integrand into a product of that and its derivative...

I don't quite see how lines 2 and 3 equate, but they seem to.

Just in case a picture helps..

... where (key in spoiler) ...

Spoiler:

Full size

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!