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

We present our recent calculation of the third order corrections to the semileptonic $b\to c$ and the muon decays. The calculation has been performed in an expansion around the limit $m_c\sim m_b$, but shows decent convergence even for $m_c=0$ from which the contribution to the muon decay can be extracted. For the semileptonic $b\to c$ decay we find large perturbative corrections in the on-shell scheme which can be significantly reduced by changing to the kinetic scheme for the bottom quark mass. These results are important input for the inclusive determination of $\mathrm{\mid }{V}_{cb}\mathrm{\mid }$ and the Fermi coupling constant $G_F$.