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How to calculate complicateda hard definite integral?

I have athe following function of 4$4$ - variables that is defined like this,

f1111[x1_, x11_, x2_, x22_] := (-(x1 - x11)^4 -   6 (x1 - x11)^2 (x2 - x22)^2  + 3 (x2 - x22)^4)/( 4 Pi 4π((x1 - x11)^2 + (x2 - x22)^2)^3)

functionthe variables are x1, x11, x2x1, x11, x2 and x22x22.

Wolfram MathematicaWolfram Mathematica can easily calculate the indefinite integral of this function.

FullSimplify[Integrate[f1111[x1FullSimplify[ Integrate[ f1111[x1, x11, x2, x22], x1, x11, x2, x22]]

The result is a verylongvery long function. However, my problem is a definite integral, which needs to be calculated in a square domain,

Integrate[   f1111[x1, x11, x2, x22],  {x1, -0.5, -0.25}, {x11, -0.25,   0}, {x2, -0.5, -0.25}, {x22, -0.5, -0.25}]

The above code is not workingdoesn't work, I waitedwaiting for about 30 minutes and nothing nothing happened.

I tried numerical integration with different methods (LocalAdaptiveLocalAdaptive and GlobalAdaptiveGlobalAdaptive), and I also tried to increase WorkingPrecisionWorkingPrecision and PrecisionGoalPrecisionGoal, but the problem of singularity and low convergence is always present.

Is there a way the get the right solution in reasnoable time, becauseIs there a way to get the correct solution in a reasonable time?

Because I will need to perform thethis integration several times.

How to calculate complicated definite integral?

I have a function of 4 variables that is defined like this,

f1111[x1_, x11_, x2_, x22_] := (-(x1 - x11)^4 -   6 (x1 - x11)^2 (x2 - x22)^2 + 3 (x2 - x22)^4)/( 4 Pi ((x1 - x11)^2 + (x2 - x22)^2)^3)

function variables are x1, x11, x2 and x22.

Wolfram Mathematica can easily calculate the indefinite integral of this function.

FullSimplify[Integrate[f1111[x1, x11, x2, x22], x1, x11, x2, x22]]

The result is a verylong function. However, my problem is definite integral, which needs to be calculated in a square domain,

Integrate[  f1111[x1, x11, x2, x22], {x1, -0.5, -0.25}, {x11, -0.25,   0}, {x2, -0.5, -0.25}, {x22, -0.5, -0.25}]

The above code is not working, I waited for about 30 minutes and nothing happened.

I tried numerical integration with different methods (LocalAdaptive and GlobalAdaptive), and I also tried to increase WorkingPrecision and PrecisionGoal, but the problem of singularity and low convergence is always present.

Is there a way the get the right solution in reasnoable time, because I will need to perform the integration several times.

How to calculate a hard definite integral?

I have the following function of $4$ - variables

f1111[x1_, x11_, x2_, x22_] := (-(x1 - x11)^4 - 6(x1 - x11)^2 (x2 - x22)^2  + 3(x2 - x22)^4)/(4π((x1 - x11)^2 + (x2 - x22)^2)^3)

the variables are x1, x11, x2 and x22.

Wolfram Mathematica can easily calculate the indefinite integral of this function.

FullSimplify[ Integrate[ f1111[x1, x11, x2, x22], x1, x11, x2, x22]]

The result is a very long function. However, my problem is a definite integral, which needs to be calculated in a square domain,

Integrate[ f1111[x1, x11, x2, x22],  {x1, -0.5, -0.25}, {x11, -0.25, 0}, {x2, -0.5, -0.25}, {x22, -0.5, -0.25}]

The above code doesn't work, waiting for about 30 minutes nothing happened.

I tried numerical integration with different methods (LocalAdaptive and GlobalAdaptive), and I also tried to increase WorkingPrecision and PrecisionGoal, but the problem of singularity and low convergence is always present.

Is there a way to get the correct solution in a reasonable time?

Because I will need to perform this integration several times.

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user57225
user57225
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How to calculate complicated definite integral?

I have a function of 4 variables that is defined like this,

f1111[x1_, x11_, x2_, x22_] := (-(x1 - x11)^4 - 6 (x1 - x11)^2 (x2 - x22)^2 + 3 (x2 - x22)^4)/( 4 Pi ((x1 - x11)^2 + (x2 - x22)^2)^3)

function variables are x1, x11, x2 and x22.

Wolfram Mathematica can easily calculate the indefinite integral of this function.

FullSimplify[Integrate[f1111[x1, x11, x2, x22], x1, x11, x2, x22]]

The result is a verylong function. However, my problem is definite integral, which needs to be calculated in a square domain,

Integrate[ f1111[x1, x11, x2, x22], {x1, -0.5, -0.25}, {x11, -0.25, 0}, {x2, -0.5, -0.25}, {x22, -0.5, -0.25}]

The above code is not working, I waited for about 30 minutes and nothing happened.

I tried numerical integration with different methods (LocalAdaptive and GlobalAdaptive), and I also tried to increase WorkingPrecision and PrecisionGoal, but the problem of singularity and low convergence is always present.

Is there a way the get the right solution in reasnoable time, because I will need to perform the integration several times.