The integration with help of Rubi gives after inserting the limits:
(1/(4*Sqrt[2]))*(Pi*Log[2] - Log[(Sqrt[2] + 1)^2]*Log[2] +
PolyLog[2, -(Sqrt[2] + 1)^2] - PolyLog[2, (Sqrt[2] - 1)^4]/2 +
Re[4*PolyLog[2, I*(Sqrt[2] - 1)] - 4*PolyLog[2, I*(Sqrt[2] + 1)] +
PolyLog[2, (Sqrt[2] + 1)^2 + I/1000000000000]] +
Im[-8*(PolyLog[2, I*(Sqrt[2] - 1)] + PolyLog[2, I*(Sqrt[2] + 1)]) +
2*PolyLog[2, (Sqrt[2] + 1)^2 + I/1000000000000]]/2)
which I don't know how to further simplify. The small imaginary offset in the argument of the PolyLog 'pushes' the sign of its imaginary part to the positive side. Otherwise the numeric result would be wrong.
Addendum: the imaginary part of the mentioned PolyLog is
2*Pi*Log[Sqrt[2] + 1]
so the result can be written
(1/(4*Sqrt[2]))*( Pi^2/3 + Pi*(2*Log[Sqrt[2] + 1] + Log[2]) -
2*Log[Sqrt[2] + 1]*(Log[Sqrt[2] + 1] + Log[2])-PolyLog[2, (Sqrt[2] - 1)^2] +
PolyLog[2, -(Sqrt[2] + 1)^2] - (1/2)*PolyLog[2, (Sqrt[2] - 1)^4] +
Re[4*PolyLog[2, I*(Sqrt[2] - 1)] - 4*PolyLog[2, I*(Sqrt[2] + 1)]]
-4*Im[PolyLog[2, I*(Sqrt[2] - 1)] + PolyLog[2, I*(Sqrt[2] + 1)]])
2.Addendum: The last may be further simplified to
(1/(4*Sqrt[2]))*(Pi^2/3 + Pi*Log[2]+2*Pi*Log[Sqrt[2]+1] -
2*Log[1+Sqrt[2]]*Log[2*(1+Sqrt[2])]-2*PolyLog[2,(Sqrt[2]-1)^2] - (Sqrt[2]+1)*
LerchPhi[-(Sqrt[2]+1)^2,2,1/2]-(Sqrt[2]-1)*LerchPhi[-(Sqrt[2]-1)^2,2, 1/2])
Integrate[]
returns the (complicated but) correct answer, so something happened in between that version and 12.2. (Someone else who can check 12 and 12.1 should add the requisite bug header to this question.) $\endgroup$int
gives the same result the the numerical one. This looks like bug in integrate. screen shot !Mathematica graphics-0.1628582917 + 0.*I
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