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As Michael has shown, semi-documented preprocessor "InterpolationPointsSubdivision" with undocumented "MaxSubregions" -> 10^9 suboption gives significant speedup in both cases. So the complete Method option should be:

As Michael has shown, semi-documented preprocessor "InterpolationPointsSubdivision" with undocumented "MaxSubregions" -> 10^9 suboption gives significant speedup in both cases. So the complete Method option should be:

It is interesting that switching off "SymbolicProcessing" suboption gives a little better performance in this case:

Method -> {"InterpolationPointsSubdivision", "MaxSubregions" -> 10^9, 
  "SymbolicProcessing" -> False}

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UPDATE 2 start

As Michael has shown, semi-documented preprocessor "InterpolationPointsSubdivision" with undocumented "MaxSubregions" -> 10^9 suboption gives significant speedup in both cases. So the complete Method option should be:

Method -> {"InterpolationPointsSubdivision", "MaxSubregions" -> 10^9, 
  Method -> {"LocalAdaptive", MaxPoints -> Infinity, 
    Method -> {"TrapezoidalRule"}}}

The rest of the answer does not include this update.

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