# Mathematica does not respect linearity of integral

Fixed in 11.3

I have the following issue, trying to evaluate an integral. Mathematica tells me

Integrate[x^2 (1 + x^2)^(1/2) + y + z, {x, y, z} ∈ Sphere[{0, 0, 0}, 1]]

(* 1/2 π (3 Sqrt - Log[2 (2 + Sqrt)]) *)


By the way, it also gives me

Integrate[x^2 (1 + x^2)^(1/2), {x, y, z} ∈ Sphere[{0, 0, 0}, 1]]

(* 1/4 π (6 Sqrt + Log[3 - 2 Sqrt]) *)


And, of course,

Integrate[y + z, {x, y, z} ∈ Sphere[{0, 0, 0}, 1]]

(* 0 *)


You can check that the two results are not equal (they differ by $\frac{3\pi}{8}\log 2$); therefore the linearity of the integral is not respected. How can I explain this fact?

Thank you in advance! This is really puzzling me up.

• Same behavior here on Mathematica 11.0.1 on macos 10.12.6. I would classify that as a bug. Note that this does not happen with NIntegrate which has some other issues instead. Nov 25, 2017 at 15:52
• Thank you for quick answer. By the way, do you know what is the bug caused by? So that one can at least try to avoid it. Nov 25, 2017 at 15:56
• I have absolutely no idea. Personally, I mostly avoid everything provided by Mathematica related to two dimensional MeshRegions; mostly for the reason that many features haven't been implemented yet. But this issue is really striking. Would you please be so kind and send a bug report to Wolfram Research? Nov 25, 2017 at 16:47
• Same behavior for 11.2 on Windows 10. Bug. Nov 25, 2017 at 21:44
• confirm it is fixed in 11.3 !Mathematica graphics Mar 10, 2018 at 21:31