The authors derive an analytical three-dimensional solution for the eigenstrain fields inside and outside an inclusion (e.g. spherical one) in a slightly cubic material. The deviation from the isotropic case is measured by a small parameter \(\mu\); expanding the corresponding integral kernel with respect to \(\mu\), all integrations can be performed analytically. Finally, the authors use discrete Fourier transforms to estimate the range of validity of the obtained closed-form solution.