# 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. – Henrik Schumacher Nov 25 '17 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. – Simone Melchiorre Chiarello Nov 25 '17 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? – Henrik Schumacher Nov 25 '17 at 16:47
• Same behavior for 11.2 on Windows 10. Bug. – bbgodfrey Nov 25 '17 at 21:44
• confirm it is fixed in 11.3 !Mathematica graphics – Nasser Mar 10 '18 at 21:31