Let \(I_{n}=\int_{0}^{1} \frac{1}{\left(1+x^{2}\right)^{n}} \mathrm{~d} x\). Prove that, for every positive integer \(n\),
\(2 n I_{n+1}=2^{-n}+(2 n-1) I_{n} .\)

Given that \(I_{1}=\frac{1}{4} \pi\), find the exact value of \(I_{3}\).