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I have to find a closed form for: $$I(a)=\int\limits_0^\infty \frac{\cos(ax)}{(1 + x^2)^3}dx.$$

I think I have to use this integral $$L(a) = \int\limits_0^\infty \frac{\cos(ax)}{1+x^2} = \frac{\pi e^{-|a|}}{2}$$

But how can I do this? How should I change variables?

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    That integral you wrote doesn't make sense, did you mean $-|a|$ instead of $|a|$? – 2017-02-27

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From the identity $$ \int_{0}^{+\infty}\frac{\cos(ax)}{1+x^2}\,dx = \frac{\pi}{2} e^{-|a|}\tag{1}$$ it also follows, through a change of variable, that for any $b>0$ $$ \int_{0}^{+\infty}\frac{\cos(ax)}{b+x^2}\,dx =\frac{\pi}{2\sqrt{b}}e^{-|a|\sqrt{b}}\tag{2}$$ holds. If we differentiate both sides of $(2)$ with respect to $b$ twice, then evaluate at $b=1$, we get: $$ 2\int_{0}^{+\infty}\frac{\cos(ax)}{(1+x^2)^3}\,dx = \frac{\pi}{8}e^{-|a|}(3+3|a|+a^2)\tag{3} $$ and we are done.