Strategy for Common Integrals, Trig or Not To Trig

The integral of $1/(1+x^2)$ is given as $tan^{-1}(x)$. I saw a derivation for it here, and I understand it, fine. But what is the strategy, rule-of-thumb that can be followed for deriving such integrals? Memorization is possible of course, I mean in case we cannot remember. How do we determine when to use trig based subsitution when not? If I see a "1 minus or plus square something" is that an indicator that I can do some trig based subsitution?

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Integrals involving quadratics or square roots of quadratics can generally be done, after completing the square, by one trig substitution or another. But keep your eyes open for easier methods. E.g., trig substitution is the hard way to do $$\int x\sqrt{1-x^2}\,dx$$
Huh? 1) Completing the square is (sometimes) part of using a trig substitution. 2) In this example, the square is already complete. 3) In this example, what you want to do is notice that $x$ is (a constant multiple of) the derivative of $1-x^2$, and that therefore you want to substitute $u=1-x^2$, $du=-2x\,dx$. This will get you there quicker than doing $x=\sin\theta$, $dx=\cos\theta\,d\theta$. –  Gerry Myerson Feb 21 '12 at 12:02