Acoustic scattering cross sections of smart furtive obstacles are studied and discussed. A smart furtive obstacle is an obstacle that, when hit by an incoming field, avoids detection through the use of a pressure current acting on its boundary. A highly parallelizable algorithm for computing the acoustic scattering cross section of smart obstacles is developed. As a case study, this algorithm is applied to the (acoustic) scattering cross section of a "smart” (furtive) simplified version of the NASA space shuttle when hit by incoming time-harmonic plane waves, the wavelengths of which are small compared to the characteristic dimensions of the shuttle. The solution to this numerically challenging scattering problem requires the solution of systems of linear equations with many unknowns and equations. Due to the sparsity of these systems of equations, they can be stored and solved using affordable computing resources. A cross section analysis of the simplified NASA space shuttle highlights three findings: i) the smart furtive obstacle reduces the magnitude of its cross section compared to the cross section of a corresponding "passive” obstacle; ii) several wave propagation directions fail to satisfactorily respond to the smart strategy of the obstacle; iii) satisfactory furtive effects along all directions may only be obtained by using a pressure current of considerable magnitude. Numerical experiments and virtual reality applications can be found at the website:              http://www.ceri.uniroma1.it/ceri/zirilli/w7.