Show that \(\frac{\mathrm{d}}{\mathrm{d} t}\left(t\left(1+t^{3}\right)^{n}\right)=(3 n+1)\left(1+t^{3}\right)^{n}-3 n\left(1+t^{3}\right)^{n-1}\).

Let \(I_{n}=\int_{0}^{1}\left(1+t^{3}\right)^{n} \mathrm{~d} t\). Using the above result, or otherwise, show that
\((3 n+1) I_{n}=2^{n}+3 n I_{n-1}\)

Hence evaluate \(I_{3}\).