Solution to Non-Isotropic Gaussian Integrals

In the course of some work, I needed to derive solutions to the following non-isotropic integral:

Where α and β are three-component vectors containing constants greater than zero.

Specifically, I required the solutions for a case of non-isotropic nuclear attraction, though I believe this integral could also arise when studying gravitational attraction. Neither Maple nor Mathematica were able to solve Eq. 1, and I haven't found it in any table of integrals. I eventually managed to derive solutions, but unfortunately the original project I needed the solutions for never panned out.

Still, these integrals seem as they may be of potential use to other researchers, so I have decided to post them here. If you find these solutions useful, please let me know in the comments. Additionally, if you use the solutions in any published work, I would appreciate if you provide credit where possible.

In the simplest case, all three elements of the α vector are identical, as are the elements of and β—i.e. we have a spherically symmetric, fully isotropic integral. In such a case, Eq. 1 collapses to the known integral

Should the system only posses cylindrical symmetry, say α_x=α_z and β_x=β_z, we obtain the solution:

Finally, should the system be fully isotropic, the following solution obtains:

Although the imaginary part of the solution in Eq. 4 should be zero, I have found that some math engines will still require you to explicitly use the Re command to isolate the real part.

Note that in the limit as χ approaches ζ, Eq. 4 collapses to Eq. 3. Similarly, in the limit as υ approaches ζ, Eq. 3 collapses to Eq. 2.