Micro-structural analyses are an important tool to understand material behavior on a macroscopic scale. The analysis of a microstructure is usually computationally very demanding and there are several reduced order modeling techniques available in literature to limit the computational costs of repetitive analyses of a single representative volume element. These techniques to speed up the integration at the micro-scale can be roughly divided into two classes; methods interpolating the integrand and cubature methods. The empirical interpolation method (high-performance reduced order modeling) and the empirical cubature method are assessed in terms of their accuracy in approximating the full-order result. A micro-structural volume element is therefore considered, subjected to four load-cases, including cyclic and path-dependent loading. The differences in approximating the micro-and macroscopic quantities of interest are highlighted, e.g. micro-fluctuations and stresses. Algorithmic speed-ups for both methods with respect to the full-order micro-structural model are quantified. The pros and cons of both classes are thereby clearly identified.