The Discontinuous Galerkin Method (DGM) stands out as a versatile method, notably, it is efficient at modeling intricate geometries since it handles non-conforming meshes natively. In particular, the Interior Penalty DG (IPDG) as proposed by Cangiani et al. [1] allows the definition of the approximated variable of interest on a bounding box encapsulating the element. Consequently, the number of Degrees Of Freedom (DOF) does not depend on the node count. So, to better capture small geometric features or curves, one can easily increase the face count of elements without changing the DOF count. Applying this approach to real engineering parts yields polyhedral elements with a substantial face count (several hundreds). Computing integrals over such complex polyhedron becomes cumbersome, especially as the integrand gets more elaborate which often happens when high order shape functions are used or when non-linearities are considered. Here performances of different integration schemes are compared, including analytical techniques only suitable for homogeneous functions [2] and numerical quadrature techniques [3, 4]. It is shown that for simple polynomial integrands, comprising of few monomials, analytical techniques are very efficient. However, as the number of monomials increases or if the integrand is non-polynomial, then resorting to numerical quadrature based techniques becomes imperative. [1] A. Cangiani, E.H. Georgoulis, P. Houston, hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Mathematical Models and Methods in Applied Sciences, Vol. 24:10, pp.2009–2041, 2014. [2] P. Antonietti, P. Houston, G. Pennesi, Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods. Journal of Scientific Computing, Vol. 77, pp. 1339–1370, 2018. [3] Y. Sudhakar, J.P. Moitinho de Almeida, W.A. Wall, An accurate, robust, and easy-toimplement method for integration over arbitrary polyhedra: Application to embedded interface methods. Journal of Computational Physics, Vol. 273, pp. 393–415, 2014. [4] A. Sommariva and M. Vianello Compression of multivariate discretemeasures and applications. Numerical Functional Analysis and Optimization, Vol. 36, pp. 1198– 1223, 2015.