Unit Disk Integral

Author

Eric W. Weisstein
January 31, 2006

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

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

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

Integrals

∫(∫_U)((((x^2)+(y^2)))^n)dydx

Integrate[Boole[x^2+y^2<1](x^2+y^2)^(n/2),{x,-∞,∞},{y,-∞,∞}]//Timing

Table[Integrate[-I ((Abs[x])^(1+n)) Beta[1-(1/(x^2)),(1/2),((2+n)/2)],{x,-1,1}],{n,0,2}]

Table[NIntegrate[-I ((Abs[x])^(1+n)) Beta[1-(1/(x^2)),(1/2),((2+n)/2)],{x,-1,1}],{n,0,5}]//Chop

-2I Integrate[(x^(2n+1))Beta[1-1/x^2,1/2,1+n],{x,0,1}]

2π Integrate[r (r^n),{r,0,1}]

2π Integrate[r (r^n),{r,0,1},Assumptions->n>-2]

Table[((2 π)/(2+n)),{n,0,5}]//N

∫(∫_U)(1/(Sqrt[(x^2)+(y^2)]))dydx

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

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

2π Integrate[1,{r,0,1}]

∫(∫_U)(e^(-1/((x^2)+(y^2)))) dydx

Integrate[Exp[-1/((x^2)+(y^2))]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

(∫_(-∞)^∞)(∫_(-∞)^∞)(e^(-|x+y|))dydx

Integrate[Exp[-Abs[x+y]]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

∫(∫_U)(e^(-((x^2)+(y^2)))) dydx

Integrate[Exp[-((x^2)+(y^2))]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]

Integrate[Exp[-((x^2)+(y^2))]Boole[x^2+y^2<1],{y,-∞,∞},{x,-∞,∞}]

NIntegrate[Exp[-((x^2)+(y^2))]Boole[x^2+y^2<1],{y,-∞,∞},{x,-∞,∞},WorkingPrecision->20]

Elementary Functions

Arg

Integrate[Arg[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

BesselJ

Integrate[BesselJ[0,x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[BesselJ[0,x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Cos

Integrate[Cos[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Cos[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Cosh

Integrate[Cosh[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Cosh[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

EllipticE

Integrate[EllipticE[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[EllipticE[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

EllipticK

Integrate[EllipticK[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[EllipticK[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Exp

Integrate[Exp[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

NIntegrate[Exp[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞},WorkingPrecision->20]//Timing

2(Exp[-x]+Exp[x])Sin[(Sqrt[1-x^2])]//FullSimplify

4Integrate[Cosh[x] Sin[(Sqrt[1-(x^2)])],{x,0,1}]

Integrate[Exp[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

NIntegrate[Exp[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞},WorkingPrecision->20]//Timing

Integrate[Exp[-(x^2+y^2)]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

NIntegrate[Exp[-(x^2+y^2)]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞},WorkingPrecision->20]//Timing

Gamma

Integrate[Gamma[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Gamma[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Identity

Integrate[Identity[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Identity[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

LambertW

Integrate[ProductLog[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[ProductLog[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Log[x+I y]

v5

Integrate[Log[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

NIntegrate[Log[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

NIntegrate[Log[x+I y]Boole[x^2+y^2<1],{x,-∞,0,∞},{y,-∞,0,∞}]//Timing

v6

Integrate[Log[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

....wrong...

NIntegrate[Log[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

NIntegrate[Log[x+I y]Boole[x^2+y^2<1],{x,-∞,0,∞},{y,-∞,0,∞}]//Timing

...wrong...

Log[x^2+y^2]

Integrate[Log[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

2π Integrate[Log[r^2]r,{r,0,1}]

4π Integrate[Log[r]r,{r,0,1}]

NIntegrate[Log[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Log Gamma

Integrate[Log[Gamma[x+I y]]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[LogGamma[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Log[Gamma[x^2+y^2]]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[LogGamma[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Log Log

Integrate[Log[Log[x+I y]]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Log[Log[x^2+y^2]]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Sin

Integrate[Sin[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Sin[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Sinh

Integrate[Sinh[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Sinh[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Tan

Integrate[Tan[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Tan[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Tanh

Integrate[Tanh[x+I y]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Tanh[x^2+y^2]Boole[x^2+y^2<1],{x,-∞,∞},{y,-∞,∞}]//Timing

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)))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

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

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

(∫_0^1)(∫_0^1)(1/((x^2) (y^2)+1))dydx

Integrate[(1/(1+(x^2)(y^2)))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[(1/(1+(x^2)(y^2)))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

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

Integrate[((-1)/((1-x y z)Log[x y]))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[((-1)/((1-x y z)Log[x y]))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

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

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

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

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

(∫_0^1)(∫_0^1)(1/(2-x y))dydx

Integrate[(1/(2-x y))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[(1/(2-x y))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

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

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

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

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

Sondow Example 3.7

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

Integrate[((-1)/((GoldenRatio-x y)Log[x y]))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[((-1)/((GoldenRatio-x y)Log[x y]))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

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

Integrate[((-1)/(((GoldenRatio^2)-x y)Log[x y]))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[((-1)/(((GoldenRatio^2)-x y)Log[x y]))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

(∫_0^1)(∫_0^1)(1/(ϕ-x y))dydx

Integrate[(1/(GoldenRatio-x y))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]

Integrate[(1/(GoldenRatio-x y))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]

(∫_0^1)(∫_0^1)(1/((ϕ^2)-x y))dydx

Integrate[(1/((GoldenRatio^2)-x y))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[(1/((GoldenRatio^2)-x y))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

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

Integrate[((-Log[x y])/((GoldenRatio^2)-x y))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[((-Log[x y])/((GoldenRatio^2)-x y))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

(∫_0^1)(∫_0^1)(1/(1-x y))dydx

Integrate[(1/(1-x y))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]

Integrate[(1/(1-x y))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]

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

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

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

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

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

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

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

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

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

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

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

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

(∫_0^1)(∫_0^1)(y/(1-(x^3) (y^3)))dydx

Integrate[(y/(1-(x^3)(y^3)))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[(y/(1-(x^3)(y^3)))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

expr=(1/(3 (Sqrt[2])))(-(Sqrt[-1-I (Sqrt[3])]) ArcTan[(1/2)-((I (Sqrt[3]))/2)]+I (Sqrt[3 (-1-I (Sqrt[3]))]) ArcTan[(1/2)-((I (Sqrt[3]))/2)]-(Sqrt[-1+I (Sqrt[3])]) ArcTan[(1/2) (1+I (Sqrt[3]))]-I (Sqrt[3 (-1+I (Sqrt[3]))]) ArcTan[(1/2) (1+I (Sqrt[3]))]+(Sqrt[-1+I (Sqrt[3])]) ArcTan[(1/(Sqrt[-(1/2) I (-I+(Sqrt[3]))]))]+I (Sqrt[3 (-1+I (Sqrt[3]))]) ArcTan[(1/(Sqrt[-(1/2) I (-I+(Sqrt[3]))]))]+(Sqrt[-1-I (Sqrt[3])]) ArcTan[(1/(Sqrt[(1/2) I (I+(Sqrt[3]))]))]-I (Sqrt[3 (-1-I (Sqrt[3]))]) ArcTan[(1/(Sqrt[(1/2) I (I+(Sqrt[3]))]))]);

FullSimplify[expr]//Timing

(∫_0^1)(∫_0^1)(1/(1-(x^2) (y^2)))dydx

Integrate[(1/(1-(x^2)(y^2)))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[(1/(1-(x^2)(y^2)))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

Example 3.16

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

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

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

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

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

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

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

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

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

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

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

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

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

Integrate[(x/(((-Log[x y]))^(3/2)))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[(x/(((-Log[x y]))^(3/2)))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

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

Integrate[(1/(((-Log[x y]))^(3/2)))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[(1/(((-Log[x y]))^(3/2)))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

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

Integrate[(1/(((-Log[x y]))^(5/4)))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[(1/(((-Log[x y]))^(5/4)))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//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))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

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

(expr=FullSimplify[(1/(4 (y^2) (-1-(y^2)+(y^4))))(2 (y^2) ((Log[-y])^2)-2 (y^4) ((Log[-y])^2)-2 (y^2) ((Log[y])^2)+2 (y^4) ((Log[y])^2)+2 (y^2) Log[-y] Log[1-(y^2)]-2 (y^4) Log[-y] Log[1-(y^2)]-2 (y^2) Log[y] Log[1-(y^2)]+2 (y^4) Log[y] Log[1-(y^2)]-2 Log[-y] Log[1-I y (Sqrt[1-(y^2)])]-2 (y^2) Log[-y] Log[1-I y (Sqrt[1-(y^2)])]+2 (y^4) Log[-y] Log[1-I y (Sqrt[1-(y^2)])]+2 Log[y] Log[1-I y (Sqrt[1-(y^2)])]+2 (y^2) Log[y] Log[1-I y (Sqrt[1-(y^2)])]-2 (y^4) Log[y] Log[1-I y (Sqrt[1-(y^2)])]-2 Log[-y] Log[1+I y (Sqrt[1-(y^2)])]-2 (y^2) Log[-y] Log[1+I y (Sqrt[1-(y^2)])]+2 (y^4) Log[-y] Log[1+I y (Sqrt[1-(y^2)])]+2 Log[y] Log[1+I y (Sqrt[1-(y^2)])]+2 (y^2) Log[y] Log[1+I y (Sqrt[1-(y^2)])]-2 (y^4) Log[y] Log[1+I y (Sqrt[1-(y^2)])]+2 (-1-(y^2)+(y^4)) PolyLog[2,-I y (Sqrt[1-(y^2)])]+2 (-1-(y^2)+(y^4)) PolyLog[2,I y (Sqrt[1-(y^2)])]+PolyLog[2,(y^2) (-1+(y^2))]+(y^2) PolyLog[2,(y^2) (-1+(y^2))]-(y^4) PolyLog[2,(y^2) (-1+(y^2))])])//Timing

Integrate[(1/(2 (y^2) (-1-(y^2)+(y^4))))((Log[-y]-Log[y]) (((y^2)-(y^4)) Log[-y]+((y^2)-(y^4)) Log[y]-(y^2) (-1+(y^2)) Log[1-(y^2)]+(-1-(y^2)+(y^4)) (Log[1-I y (Sqrt[1-(y^2)])]+Log[1+I y (Sqrt[1-(y^2)])]))),{y,-1,1}]//Timing

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

Integrate[(((Log[x y])^4)/(((1+x y))^2))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[(((Log[x y])^4)/(((1+x y))^2))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing

Answer:

NIntegrate[(((Log[x y])^4)/(((1+x y))^2))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//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]))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

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

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

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

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

(∫_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]))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

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

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

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

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

(∫_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)))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

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

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

Integrate[((1-x)/(((-Log[x y]))^(5/2)))Boole[x^2+y^2<1],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Timing

Integrate[((1-x)/(((-Log[x y]))^(5/2)))Boole[x^2+y^2<1],{y,-Infinity,Infinity},{x,-Infinity,Infinity}]//Timing