performance tuning - How to avoid repetitive calculation when doing numerical integral?

Suppose I have a function f[x] which is very complicated, together with a function g[f[x]]+h[x] to integrate. That is:

NIntegrate[{f[x], g[f[x]]+h[x]}, {x,0,1}]

I suppose mathematica will calculate f[x] and g[f[x]]+h[x] seperately, thus calculated f[x] twice. How can I speed up the calculation by telling mathematica calculate f[x] only once?

A concrete example suggested by the comments:

ClearAll["Global`*"]

f[x_, y_, z_] := Exp[Sin[x]] + Cos[y + z];


NIntegrate[{f[x, y, z], Sqrt[f[x, y, z]] + x}, {x, 0, 10}, {y, 0, 10}, {z, 0, 10}, 
Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}, 
PrecisionGoal -> 7] // Timing

{8.85938, {1429.54, 6081.95 + 59.0571 I}}

g[x_, y_, z_] := g[x, y, z] = Exp[Sin[x]] + Cos[y + z];

NIntegrate[{g[x, y, z], Sqrt[g[x, y, z]] + x}, {x, 0, 10}, {y, 0, 10}, {z, 0, 10}, 

Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}, 
PrecisionGoal -> 7] // Timing

{8.90625, {1429.54, 6081.95 + 59.0571 I}}

In this example the memoization seems not speed up the calcuation...

Comments

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