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How does Mathematica integrate?

Mathematica can perform the following integral with apparent ease (returning very fast with the result)

Integrate[(1 - a Tan[b x]^2)^(-1/2), x]

The answer happens to be

(ArcTan[(Sqrt[2] Sqrt[1 + a] Sin[b x])/Sqrt[  1 - a + (1 + a) Cos[2 b x]]] Sqrt[1 - a + (1 + a) Cos[2 b x]]   Sec[b x])/(Sqrt[2] Sqrt[1 + a] b Sqrt[1 - a Tan[b x]^2])

Is there a way to have Mathematica show what sort of substitutions it made along the way?

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Peeter Joot
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    Even if this is possible somehow, given the way Mathematica performs integration, those are not the substitutions you are looking for :). Mathematica does not do the integration as humans do, it first reduces the integral to the most general form possible, and then tries to simplify that. You might have better luck with WolframAlpha which can sometimes show the steps. – Leonid Shifrin Jan 20 '13 at 17:39
  • I'd seen the 'How does Mathematica integrate' and didn't see what I wanted in there. – Peeter Joot Jan 20 '13 at 18:04

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