Let

\[ I=\int_1^3\frac{x^3}{3+x^2}\,dx. \]

(a) Using the substitution \(x=\sqrt3\tan u\), show that \(I=\int_{\frac16\pi}^{\frac13\pi}3\tan^3u\,du\).

(b) Hence, or otherwise, find the exact value of \(I\). Give your answer in the form \(p+q\ln r\), where \(p\), \(q\) and \(r\) are rational.