  The current discrepancy of theory and experiment observed recently in muonic
hydrogen necessitates a reinvestigation of all corrections to contribute to the
Lamb shift in muonic hydrogen muH, muonic deuterium muD, the muonic 3He ion, as
well as in the muonic 4He ion. Here, we choose a semi-analytic approach and
evaluate a number of higher-order corrections to vacuum polarization (VP)
semi-analytically, while remaining integrals over the spectral density of VP
are performed numerically. We obtain semi-analytic results for the second-order
correction, and for the relativistic correction to VP. The self-energy
correction to VP is calculated, including the perturbations of the Bethe
logarithms by vacuum polarization. Subleading logarithmic terms in the
radiative-recoil correction to the 2S-2P Lamb shift of order alpha (Zalpha)^5
mu^3 ln(Zalpha)/(m_mu m_N) are also obtained. All calculations are
nonperturbative in the mass ratio of orbiting particle and nucleus.
