Unit Square Integral

Author

Eric W. Weisstein
February 20, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/UnitSquareIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/UnitSquareIntegral.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Code

Integrate1D[f_,{x_,0,1},{y_,0,1}]:=Integrate[Integrate[(f/Y)/.{x->X/Y,y->Y},{Y,X,1},Assumptions->0<X<1],{X,0,1}]

Abs Sum and Difference

(∫_0^1)(∫_0^1)((|x-y|)^n)dydx

Integrate[((Abs[x-y])^n),{x,0,1},{y,0,1},Assumptions->n>-1]//Factor

Table[(2/((1+n) (2+n))),{n,0,5}]

Table[Integrate[((Abs[x-y])^n),{x,0,1},{y,0,1}],{n,0,5}]

(∫_0^1)(∫_0^1)((|x+y|)^n)dydx

Integrate[((Abs[x+y])^n),{x,0,1},{y,0,1},Assumptions->n>-2]//Factor

Table[Integrate[((Abs[x+y])^n),{x,0,1},{y,0,1}],{n,0,5}]

Table[((2 (-1+(2^(1+n))))/((1+n) (2+n))),{n,0,5}]

Integrals

(∫_0^1)(∫_0^1)(1/((x^2)+(y^2)))dydx

Integrate[(1/((x^2+y^2))),{x,0,1},{y,0,1}]

Integrate[(1/((x^2+y^2))),{x,0,1},{y,0,1},GenerateConditions->False]

Integrate[(1/((x^2)+(y^2))),{x,ϵ,1},{y,ϵ,1},Assumptions->0<ϵ<1]

Limit[%,ϵ->0]

Exact Hadamard-regularized Result

<<Graphics`InequalityGraphics`

InequalityPlot[x^2+y^2>1&&0<x<1&&0<y<1,{x},{y}]

Integrate[Boole[x^2+y^2>1&&0<x<1&&0<y<1]/(x^2+y^2),{x,-∞,∞},{y,-∞,∞}]

4.2

Wrong

Integrate[1/(x^2+y^2),{x,0,1},{y,0,1}]

N[%]

i=Integrate[(1/((x^2)+(y^2))),{x,ϵ,1},{y,ϵ,1}]

Simplify[i/.ϵ->1]

{Limit[PolyLog[2,I ϵ],ϵ->0],Limit[PolyLog[2,-I ϵ],ϵ->0]}

FullSimplify[(1/2) (-4 Catalan+I PolyLog[2,-(I/ϵ)]-I PolyLog[2,(I/ϵ)])]

Limit[I PolyLog[2,-(I/ϵ)]-I PolyLog[2,(I/ϵ)],ϵ->0]

N[Table[I PolyLog[2,-(I/(10^(-n)))]-I PolyLog[2,(I/(10^(-n)))],{n,30}],20]

5.0

Hadamard-Regularized

(∫_0^1)(∫_0^1)((Boole[(x^2)+(y^2)>1])/((x^2)+(y^2)))dydx

Integrate[Boole[x^2+y^2>1]/(x^2+y^2),{x,0,1},{y,0,1}]

Integrate[Boole[x^2+y^2>1]/(x^2+y^2),{x,0,1},{y,0,1},Assumptions->0<x<1&&0<y<1]

FullSimplify[((ArcCot[(Sqrt[-1+(1/(x^2))])]-ArcTan[x])/x),0<x<1]

v6.0

Divergent

Integrate[(1/((x^2)+(y^2))),{x,0,1},{y,0,1}]

Hadamard-Regularized

Integrate[Boole[x^2+y^2>1]/(x^2+y^2),{x,0,1},{y,0,1}]

N[%]

(∫_0^1)(∫_0^1)(1/((x^2)+(y^2)+1))dydx

Integrate[(1/((x^2)+(y^2)+1)),{x,0,1},{y,0,1}]

Integrate1D[(1/((x^2)+(y^2)+1)),{x,0,1},{y,0,1}]//Timing

Integrate[(Y/((X^2)+(Y^2)+(Y^4))),{Y,X,1},GenerateConditions->False]

NIntegrate[((ArcTan[(3/(Sqrt[-1+4 (X^2)]))]-ArcTan[((1+2 (X^2))/(Sqrt[-1+4 (X^2)]))])/(Sqrt[-1+4 (X^2)])),{X,0,1}]

Integrate[((ArcTan[(3/(Sqrt[-1+4 (X^2)]))]-ArcTan[((1+2 (X^2))/(Sqrt[-1+4 (X^2)]))])/(Sqrt[-1+4 (X^2)])),{X,0,1}]//Timing

Integrate[((ArcTan[(3/(Sqrt[-1+4 (X^2)]))]-ArcTan[((1+2 (X^2))/(Sqrt[-1+4 (X^2)]))])/(Sqrt[-1+4 (X^2)])),X]

Integrate[((ArcTan[(3/(Sqrt[-1+4 (X^2)]))])/(Sqrt[-1+4 (X^2)])),{X,0,1}]//Timing

Integrate[((ArcTan[((1+2 (X^2))/(Sqrt[-1+4 (X^2)]))])/(Sqrt[-1+4 (X^2)])),{X,0,1}]//Timing

NIntegrate[((ArcTan[Cos[θ]])/(Cos[θ])),{θ,0,π/4},WorkingPrecision->20]

NIntegrate[(1/((x^2)+(y^2)+1)),{x,0,1},{y,0,1},WorkingPrecision->20]

N[-Catalan + (Pi*ArcSinh[1])/2 + HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6,20]

(∫_0^1)(∫_0^1)(1/(Sqrt[(x^2)+(y^2)+1]))dydx

Integrate[(1/(Sqrt[(x^2)+(y^2)+1])),{x,0,1},{y,0,1}]

Guillera and Sondow Integrals

(∫_0^1)(∫_0^1)((((x^(u-1)) (y^(v-1))) (((-log(x y)))^s))/(1-x y z))dydx

Integrate[(((x^(u-1))(y^(v-1)))/(1-x y z))(((-Log[x y]))^s),{x,0,1},{y,0,1}]//Timing

Integrate1D[(((x^(u-1))(y^(v-1)))/(1-x y z))(((-Log[x y]))^s),{x,0,1},{y,0,1}]//Timing

Answer:

(∫_0^1)(∫_0^1)((((-log(x y)))^(n-1))/(1-x y z))dydx

Integrate[((((-Log[x y]))^(n-1))/(1-x y z)),{x,0,1},{y,0,1}]//Timing

Integrate1D[((((-Log[x y]))^(n-1))/(1-x y z)),{x,0,1},{y,0,1}]//Timing

Integrate[((((-Log[x y]))^(n-1))/(1-x y z)),{y,0,1},{x,0,1}]//Timing

Answer:

(∫_0^1)(∫_0^1)-(1/((1-x y z) log(x y)))dydx

Integrate[((-1)/((1-x y z)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-1)/((1-x y z)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate[((-1)/((1-x y z)Log[x y])),{y,0,1},{x,0,1}]//Timing

Answer:

(∫_0^1)(∫_0^1)(((((x y))^(u-1)) (((-log(x y)))^s))/(1-x y z))dydx

Integrate[((((x y))^(u-1))/(1-x y z))(((-Log[x y]))^s),{x,0,1},{y,0,1}]//Timing

Integrate1D[((((x y))^(u-1))/(1-x y z))(((-Log[x y]))^s),{x,0,1},{y,0,1}]//Timing

Integrate[((((x y))^(u-1))/(1-x y z))(((-Log[x y]))^s),{y,0,1},{x,0,1}]//Timing

Answer:

Guillera and Sondow Integrals

(∫_0^1)(∫_0^1)-((x log(x y))/((x^2) (y^2)+1))dydx

Integrate[((-x Log[x y])/(1+(x^2)(y^2))),{x,0,1},{y,0,1}]//Timing

Integrate[((-x Log[x y])/(1+(x^2)(y^2))),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)-((x log(x y))/(1-(x^2) (y^2)))dydx

Integrate[((-x Log[x y])/(1-(x^2)(y^2))),{x,0,1},{y,0,1}]//Timing

Integrate[((-x Log[x y])/(1-(x^2)(y^2))),{y,0,1},{x,0,1}]//Timing

Integrate1D[((-x Log[x y])/(1-(x^2)(y^2))),{x,0,1},{y,0,1}]//Timing

(∫_0^1)(∫_0^1)-((log(x y))/((x^2) (y^2)+1))dydx

Integrate[((-Log[x y])/(1+(x^2)(y^2))),{x,0,1},{y,0,1}]//Timing

Integrate[((-Log[x y])/(1+(x^2)(y^2))),{y,0,1},{x,0,1}]//Timing

Integrate1D[((-Log[x y])/(1+(x^2)(y^2))),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)(1/(i x y+1))dydx

Integrate[(1/(1+x y I)),{x,0,1},{y,0,1}]//Timing

Integrate[(1/(1+x y I)),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)(1/((x^2) (y^2)+1))dydx

Integrate[(1/(1+(x^2)(y^2))),{x,0,1},{y,0,1}]//Timing

Integrate[(1/(1+(x^2)(y^2))),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)-(1/((2-x y) log(x y)))dxdy

Integrate[((-1)/((2-x y)Log[x y])),{y,0,1},{x,0,1}]//Timing

Integrate[((-1)/((2-x y)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-1)/((2-x y)Log[x y])),{x,0,1},{y,0,1}]

Integrate1D[((-1)/((2-x y)Log[x y])),{y,0,1},{x,0,1}]

Answer:

NIntegrate[((-1)/((2-x y)Log[x y])),{x,0,1},{y,0,1}]//Timing

(∫_0^1)(∫_0^1)(1/(2-x y))dydx

Integrate[(1/(2-x y)),{x,0,1},{y,0,1}]//Timing

Integrate[(1/(2-x y)),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)-((log(x y))/(2-x y))dydx

Integrate[((-Log[x y])/(2-x y)),{x,0,1},{y,0,1}]//Timing

Integrate[((-Log[x y])/(2-x y)),{y,0,1},{x,0,1}]//Timing

Sondow Example 3.6

(∫_0^1)(∫_0^1)-(x/((2-x y) log(x y)))dydx

Integrate[((-x)/((2-x y)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-x)/((2-x y)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate[((-x)/((2-x y)Log[x y])),{y,0,1},{x,0,1}]//Timing

Integrate1D[((-x)/((2-x y)Log[x y])),{y,0,1},{x,0,1}]//Timing

Simplify[(1/((-2+X) Log[X]))-(X/((-2+X) Log[X]))]

Answer:

NIntegrate[((-x)/((2-x y)Log[x y])),{y,0,1},{x,0,1}]

-Derivative[1,0][PolyLog][0,1/2]//N

Sondow Example 3.7

(∫_0^1)(∫_0^1)-(1/((ϕ-x y) log(x y)))dydx

Integrate[((-1)/((GoldenRatio-x y)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-1)/((GoldenRatio-x y)Log[x y])),{x,0,1},{y,0,1}]//FullSimplify//Timing

Integrate[((-1)/((GoldenRatio-x y)Log[x y])),{y,0,1},{x,0,1}]//Timing

Integrate1D[((-1)/((GoldenRatio-x y)Log[x y])),{y,0,1},{x,0,1}]//Timing

Answer:

NIntegrate[((-1)/((GoldenRatio-x y)Log[x y])),{x,0,1},{y,0,1}]

2Log[GoldenRatio]//N

(∫_0^1)(∫_0^1)-(1/(((ϕ^2)-x y) log(x y)))dydx

Integrate[((-1)/(((GoldenRatio^2)-x y)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-1)/(((GoldenRatio^2)-x y)Log[x y])),{x,0,1},{y,0,1}]//FullSimplify//Timing

Integrate[((-1)/(((GoldenRatio^2)-x y)Log[x y])),{y,0,1},{x,0,1}]//Timing

Integrate1D[((-1)/(((GoldenRatio^2)-x y)Log[x y])),{y,0,1},{x,0,1}]//FullSimplify//Timing

Answer:

NIntegrate[((-1)/(((GoldenRatio^2)-x y)Log[x y])),{x,0,1},{y,0,1}]

Log[GoldenRatio]//N

(∫_0^1)(∫_0^1)(1/(ϕ-x y))dydx

Integrate[(1/(GoldenRatio-x y)),{x,0,1},{y,0,1}]

Integrate[(1/(GoldenRatio-x y)),{y,0,1},{x,0,1}]

FunctionExpand[PolyLog[2,(2/(1+(Sqrt[5])))]]

N[%]

((π^2)/10)-Log[GoldenRatio]^2//N

(∫_0^1)(∫_0^1)(1/((ϕ^2)-x y))dydx

Integrate[(1/((GoldenRatio^2)-x y)),{x,0,1},{y,0,1}]//Timing

Integrate[(1/((GoldenRatio^2)-x y)),{y,0,1},{x,0,1}]//Timing

N[PolyLog[2,(2/(3+(Sqrt[5])))]]

(∫_0^1)(∫_0^1)-((log(x y))/((ϕ^2)-x y))dydx

Integrate[((-Log[x y])/((GoldenRatio^2)-x y)),{x,0,1},{y,0,1}]//Timing

Integrate[((-Log[x y])/((GoldenRatio^2)-x y)),{y,0,1},{x,0,1}]//Timing

Answer:

FullSimplify[2 PolyLog[3,(1/2) (3-(Sqrt[5]))]]

NIntegrate[((-Log[x y])/((GoldenRatio^2)-x y)),{x,0,1},{y,0,1}]

((8Zeta[3])/5)-((4(π^2)Log[GoldenRatio])/15)+((4((Log[GoldenRatio])^3))/3)//N

FullSimplify[Subtract@@{2 PolyLog[3,(1/2) (3-(Sqrt[5]))],((8Zeta[3])/5)-((4(π^2)Log[GoldenRatio])/15)+((4((Log[GoldenRatio])^3))/3)}]

Developer`ZeroQ[%]//Timing

(∫_0^1)(∫_0^1)(1/(1-x y))dydx

Integrate[(1/(1-x y)),{x,0,1},{y,0,1}]

Integrate[(1/(1-x y)),{y,0,1},{x,0,1}]

(∫_0^1)(∫_0^1)-((log(x y))/(1-x y))dydx

Integrate[((-Log[x y])/(1-x y)),{x,0,1},{y,0,1}]//Timing

Integrate[((-Log[x y])/(1-x y)),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)-(x/((x y+1) log(x y)))dydx

Integrate[((-x)/((1+x y)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-x)/((1+x y)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate[((-x)/((1+x y)Log[x y])),{y,0,1},{x,0,1}]//Timing

Integrate1D[((-x)/((1+x y)Log[x y])),{y,0,1},{x,0,1}]//Timing

Answer:

NIntegrate[((-x)/((1+x y)Log[x y])),{x,0,1},{y,0,1}]

Log[(π/2)]//N

(∫_0^1)(∫_0^1)-(x/(((x^2) (y^2)+1) log(x y)))dydx

Integrate[((-x)/((1+(x^2)(y^2))Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-x)/((1+(x^2)(y^2))Log[x y])),{x,0,1},{y,0,1}]//FullSimplify//Timing

Integrate[((-x)/((1+(x^2)(y^2))Log[x y])),{y,0,1},{x,0,1}]//Timing

Integrate1D[((-x)/((1+(x^2)(y^2))Log[x y])),{y,0,1},{x,0,1}]//FullSimplify//Timing

NIntegrate[((-x)/((1+(x^2)(y^2))Log[x y])),{x,0,1},{y,0,1}]

Log[((Sqrt[2π])/((Gamma[(3/4)])^2))]//N

(∫_0^1)(∫_0^1)-(1/(((x^2) (y^2)+1) log(x y)))dydx

Integrate[((-1)/((1+(x^2)(y^2))Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-1)/((1+(x^2)(y^2))Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate[((-1)/((1+(x^2)(y^2))Log[x y])),{y,0,1},{x,0,1}]//Timing

NIntegrate[((-1)/((1+(x^2)(y^2))Log[x y])),{x,0,1},{y,0,1}]

(Pi/4)//N

(∫_0^1)(∫_0^1)(y/(1-(x^3) (y^3)))dydx

Integrate[(y/(1-(x^3)(y^3))),{x,0,1},{y,0,1}]//Timing

(∫_0^1)(∫_0^1)(1/(1-(x^2) (y^2)))dydx

Integrate[(1/(1-(x^2)(y^2))),{x,0,1},{y,0,1}]//Timing

Integrate[(1/(1-(x^2)(y^2))),{y,0,1},{x,0,1}]//Timing

Example 3.16

(∫_0^1)(∫_0^1)((log(2-x))/(1-x y))dydx

Integrate[((Log[2-x])/(1-x y)),{x,0,1},{y,0,1}]//Timing

Integrate[((Log[2-x])/(1-x y)),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)((log(2-x y))/(1-x y))dydx

Integrate[((Log[2-x y])/(1-x y)),{x,0,1},{y,0,1}]//Timing

Integrate[((Log[2-x y])/(1-x y)),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)-((log(1-x))/(1-x y))dydx

Integrate[((-Log[1-x])/(1-x y)),{x,0,1},{y,0,1}]//Timing

Integrate[((-Log[1-x])/(1-x y)),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)-((log(1-x y))/(1-x y))dydx

Integrate[((-Log[1-x y])/(1-x y)),{x,0,1},{y,0,1}]//Timing

Integrate[((-Log[1-x y])/(1-x y)),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)(x/(((-log(x y)))^(3/2)))dydx

Integrate[(x/(((-Log[x y]))^(3/2))),{x,0,1},{y,0,1}]//Timing

Integrate[(x/(((-Log[x y]))^(3/2))),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)(1/(((-log(x y)))^(3/2)))dydx

Integrate[(1/(((-Log[x y]))^(3/2))),{x,0,1},{y,0,1}]//Timing

Integrate[(1/(((-Log[x y]))^(3/2))),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)(1/(((-log(x y)))^(5/4)))dydx

Integrate[(1/(((-Log[x y]))^(5/4))),{x,0,1},{y,0,1}]//Timing

Integrate[(1/(((-Log[x y]))^(5/4))),{y,0,1},{x,0,1}]//Timing

(∫_0^1)(∫_0^1)((x (log^2)(x y))/((((x^2) (y^2)+1))^2))dydx

Integrate[((x ((Log[x y])^2))/(((1+(x^2)(y^2)))^2)),{x,0,1},{y,0,1}]//Timing

Integrate1D[((x ((Log[x y])^2))/(((1+(x^2)(y^2)))^2)),{x,0,1},{y,0,1}]//Timing

Integrate[((x ((Log[x y])^2))/(((1+(x^2)(y^2)))^2)),{y,0,1},{x,0,1}]//Timing

Integrate1D[((x ((Log[x y])^2))/(((1+(x^2)(y^2)))^2)),{y,0,1},{x,0,1}]//Timing

Expand[Catalan+(1/96) (π^2) (-2+3 π)]

(∫_0^1)(∫_0^1)(((log^4)(x y))/(((x y+1))^2))dydx

Integrate[(((Log[x y])^4)/(((1+x y))^2)),{x,0,1},{y,0,1}]//Timing

Integrate1D[(((Log[x y])^4)/(((1+x y))^2)),{x,0,1},{y,0,1}]//Timing

Integrate[(((Log[x y])^4)/(((1+x y))^2)),{y,0,1},{x,0,1}]//Timing

Integrate1D[(((Log[x y])^4)/(((1+x y))^2)),{y,0,1},{x,0,1}]//Timing

Answer:

NIntegrate[(((Log[x y])^4)/(((1+x y))^2)),{y,0,1},{x,0,1}]//Timing

(225/2)Zeta[5]//N

(∫_0^1)(∫_0^1)-(1/(((((x^2) (y^2)+1))^2) log(x y)))dydx

Integrate[((-1)/((((1+(x^2)(y^2)))^2)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-1)/((((1+(x^2)(y^2)))^2)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate[((-1)/((((1+(x^2)(y^2)))^2)Log[x y])),{y,0,1},{x,0,1}]//Timing

Answer:

NIntegrate[((-1)/((((1+(x^2)(y^2)))^2)Log[x y])),{x,0,1},{y,0,1}]

((π+2)/8)//N

(∫_0^1)(∫_0^1)-(x/((((x y+1))^2) log(x y)))dydx

Integrate[((-x)/((((1+x y))^2)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-x)/((((1+x y))^2)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate[((-x)/((((1+x y))^2)Log[x y])),{y,0,1},{x,0,1}]//Timing

Answer:

NIntegrate[((-x)/((((1+x y))^2)Log[x y])),{y,0,1},{x,0,1}]//Timing

Log[((Glaisher^6)/((2^(1/6))(Sqrt[π E])))]//N

(∫_0^1)(∫_0^1)-((x^2)/(((((x^2) (y^2)+1))^2) log(x y)))dxdy

Integrate[((-(x^2))/((((1+(x^2)(y^2)))^2)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate1D[((-(x^2))/((((1+(x^2)(y^2)))^2)Log[x y])),{x,0,1},{y,0,1}]//Timing

Integrate[((-(x^2))/((((1+(x^2)(y^2)))^2)Log[x y])),{y,0,1},{x,0,1}]//Timing

Answer:

NIntegrate[((-(x^2))/((((1+(x^2)(y^2)))^2)Log[x y])),{x,0,1},{y,0,1}]

(Catalan/π)//N

(∫_0^1)(∫_0^1)((1-x)/((1-x y) (-log(x y))))dxdy

Integrate[((1-x)/((1-x y)(-Log[x y]))),{x,0,1},{y,0,1}]//Timing

Integrate1D[((1-x)/((1-x y)(-Log[x y]))),{x,0,1},{y,0,1}]//Timing

Integrate[((1-x)/((1-x y)(-Log[x y]))),{y,0,1},{x,0,1}]//Timing

Answer:

NIntegrate[((1-x)/((1-x y)(-Log[x y]))),{y,0,1},{x,0,1}]//Timing

EulerGamma//N

(∫_0^1)(∫_0^1)((1-(x^2))/(((x^2) (y^2)+1) (log^2)(x y)))dxdy

Integrate[((1-(x^2))/((1+(x^2) (y^2))((Log[x y])^2))),{x,0,1},{y,0,1}]//Timing

Integrate1D[((1-(x^2))/((1+(x^2) (y^2))((Log[x y])^2))),{x,0,1},{y,0,1}]//Timing

FullSimplify[((-1+(X^2)-2 Log[X])/(2 (1+(X^2)) ((Log[X])^2)))]

Integrate[((1-(x^2))/((1+(x^2) (y^2))((Log[x y])^2))),{y,0,1},{x,0,1}]//Timing

Answer:

NIntegrate[((1-(x^2))/((1+(x^2) (y^2))((Log[x y])^2))),{x,0,1},{y,0,1}]

Log[((Gamma[(1/4)])/(2Gamma[(3/4)]))]+((2Catalan)/π)-(1/2)//N

(∫_0^1)(∫_0^1)((1-x)/(((-log(x y)))^(5/2)))dydx

Integrate[((1-x)/(((-Log[x y]))^(5/2))),{x,0,1},{y,0,1}]//Timing

Integrate[((1-x)/(((-Log[x y]))^(5/2))),{y,0,1},{x,0,1}]//Timing