I tried to show that $$\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$$ using contour integration
so $$\int_{C} \frac{dz}{(z^2+1)^{n+1}}=\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}} +\int_{\gamma}..=2\pi i\frac{1}{n!}Res_{z=i}\frac{1}{(z^2+1)^{n+1}}$$
$$\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}} +0=2\pi i\frac{1}{n!}Res_{z=i}\frac{1}{(z^2+1)^{n+1}}$$
$$\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac{2\pi i}{n!}(\frac {d^n}{dz^n}(z+i)^{-n-1})_{z=i}$$ $$\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac{2\pi i}{n!}(-1)^np(n+1,n)(2i)^{-2n-1}$$ $$\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac{\pi }{2^n}$$
where is $p(n,r)=(n)(n-1)......(n-r+1)$
so where is the mistake !
