Hint 6

In the coordinate system defined by the principal axes, you should find that the rotational kinetic energy $T={1\over 2} \hbox{{\boldmath$\omega$ }}\cdot{\bf L}$ takes the simple form

$$T={1\over 2}\Bigl(I_1\omega_1^2+I_2\omega_2^2+I_3\omega_3^2\Bigr).$$

Differentiate this with respect to $t$ and use Euler's equations to substitute for $I_1\dot\omega_1$ etc.