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We present our recent calculation of the third order corrections to thesemileptonic $b \to c$ and the muon decays. The calculation has been performedin an expansion around the limit $m_c \sim m_b$, but shows decent convergenceeven for $m_c=0$ from which the contribution to the muon decay can beextracted. For the semileptonic $b \to c$ decay we find large perturbativecorrections in the on-shell scheme which can be significantly reduced bychanging to the kinetic scheme for the bottom quark mass. These results areimportant input for the inclusive determination of $|V_{cb}|$ and the Fermicoupling constant $G_F$.

$\mathcal{O}(\alpha_s^3)$ corrections to semileptonic $b \to c$ decays in the heavy daughter approximation