Let \(I_{n}=\int_{0}^{1} \frac{1}{\left(1+x^{4}\right)^{n}} \mathrm{~d} x\). By considering \(\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x}{\left(1+x^{4}\right)^{n}}\right)\), show that
\(4 n I_{n+1}=\frac{1}{2^{n}}+(4 n-1) I_{n} .\)

Given that \(I_{1}=0.86697\), correct to 5 decimal places, find \(I_{3}\).