Complex analysis on real integral Use complex analysis to compute the real integral
$$\int_{-\infty}^\infty \frac{dx}{(1+x^2)^3}$$
Attempt
I think I want to consider this as the real part of
$$\int_{-\infty}^\infty \frac{dz}{(1+z^2)^3}$$
and then apply the residue theorem. However, I am not sure how that is the complex form and the upper integral is the real part and how to apply.

It is much faster to apply Feynman's trick. For any $a>0$ we clearly have
$$ \int_{-\infty}^{+\infty}\frac{dx}{a+x^2} = \frac{\pi}{\sqrt{a}} \tag{1} $$
and by applying $\frac{d^2}{da^2}$ to both sides:
$$ \int_{-\infty}^{+\infty}\frac{2\,dx}{(a+x^2)^3} = \frac{3\pi}{4a^2\sqrt{a}} \tag{2} $$
hence by evaluating $(2)$ at $a=1$ we instantly get
$$ \int_{-\infty}^{+\infty}\frac{dx}{(1+x^2)^3}\,dx = \frac{3\pi}{8}\tag{3}$$
and in general
$$ \forall m\geq 1,\qquad \int_{-\infty}^{+\infty}\frac{dx}{(1+x^2)^{m}}\,dx = \frac{\pi}{4^{m-1}}\binom{2m-2}{m-1}.\tag{4}$$
