  We calculate the fully differential medium-induced radiative spectrum at
next-to-leading order (NLO) accuracy within the Improved Opacity Expansion
(IOE) framework. This scheme allows us to gain analytical control of the
radiative spectrum at low and high gluon frequencies simultaneously. The high
frequency regime can be obtained in the standard opacity expansion framework in
which the resulting power series diverges at the characteristic frequency
$\omega_c\sim \hat q L^2$. In the IOE, all orders in opacity are resumed
systematically below $\omega_c$ yielding an asymptotic series controlled by
logarithmically suppressed remainders down to the thermal scale $T \ll
\omega_c$, while matching the opacity expansion at high frequency. Furthermore,
we demonstrate that the IOE at NLO accuracy reproduces the characteristic
Coulomb tail of the single hard scattering contribution as well as the Gaussian
distribution resulting from multiple soft momentum exchanges. Finally, we
compare our analytic scheme with a recent numerical solution, that includes a
full resummation of multiple scatterings, for LHC-inspired medium parameters.
We find a very good agreement both at low and high frequencies showcasing the
performance of the IOE which provides for the first time accurate analytic
formulas for radiative energy loss in the relevant perturbative kinematic
regimes for dense media.
