Triangle Triangle Picking

Author

Eric W. Weisstein
August 6, 2010

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComputationalGeometry/TriangleTrianglePicking.nb.

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

©2010 Wolfram Research, Inc. except for portions noted otherwise

Initialization

Animation

RandomTriangles[v_List,n_]:=Table[{{Hue[Random[]],Polygon[#]},Line[Append[#,First[#]]]}&@RandomTrianglePoints[v,3],{n}]

l=RandomTriangles[Rest[v],50];
movie=Table[Graphics[
{Triangle[v],Take[l,n]},
AspectRatio->Automatic],{n,0,Length[l]}];

Export[gifs<>"tritripk.gif",Show[#,PlotRange->{{0,1},{0,Sqrt[3]/2}}]&/@movie]//Timing

Show[Graphics[{Triangle[v],l},AspectRatio->Automatic]]

Numeric

StandardDeviationOfMean[l_]:=Sqrt[Variance[l]/Length[l]]
StandardDeviationOfVariance[l_]:=Module[{n=Length[l]},
Sqrt[(n-1)^2/n^3(Plus@@((l-Mean[l])^4)/n)-(n-1)(n-3)Variance[l]^2/n^3]
]

<<DiscreteMath`ComputationalGeometry`

NonRandomTrianglePoints[v_List,n_]:=Module[{},
Table[#1v[[1]]+(1-#1)#2v[[2]]&@@Table[Random[],{2}],{n}]
]

RandomQuadrilateralPoints[v_List,n_]:=Module[{},
Table[#1v[[1]]+#2v[[2]]&@@Table[Random[],{2}],{n}]
]

RandomTrianglePoints[v_List,n_]:=Select[RandomQuadrilateralPoints[v,n],Length[ConvexHull[Join[v,{{0,0},#}]]]==3&]

RandomTrianglePoints[v_List,n_]:=Module[{p,q=Plus@@v},
Table[If[Length[ConvexHull[Join[v,{{0,0},p=RandomQuadrilateralPoints[v,1][[1]]}]]]==3,p,q-p],{n}]
]

TriangleArea[v_List?MatrixQ]:=Abs[Det[Append[#,1]&/@v]]/2/;Dimensions[v]=={3,2}

AbsSq[x_]:=Sqrt[x^2]

f[x1_,y1_,x2_,y2_,x3_,y3_?NumericQ]:=TriangleArea[{{x1,y1},{x2,y2},{x3,y3}}]

16NIntegrate[f[x1,y1,x2,y2,x3,y3],{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3},Method->QuasiMonteCarlo]

16NIntegrate[f[x1,y1,x2,y2,x3,y3],{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3},Method->QuasiMonteCarlo,PrecisionGoal->4,MaxPoints->10^6]//Timing

N[1/12,10]

Monte Carlo

TriangleArea[{{x1_,y1_},{x2_,y2_},{x3_,y3_}}]:=Abs[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3]/2

Right Isosceles

Triangle with edge length 1

Length[pts=Partition[Sort/@Table[RandomReal[],{10^7},{2}],3]]//Timing

Length[pts=Partition[Select[Table[{RandomReal[{minx,maxx}],RandomReal[{miny,maxy}]},{10^5}],f@@#&],3]]//Timing

Graphics[Transpose[{{Red,Green,Blue},Point/@Transpose[Take[pts,10^3]]}],Axes->True,AxesOrigin->{0,0}]

areas=TriangleArea/@pts;//Timing

Partition[Through[{Mean,StandardDeviationOfMean,Variance,StandardDeviationOfVariance}[areas]],2]

Through[{Min,Mean,Variance,Max}[areas]]

1/12./2

Histogram[areas,AxesLabel->{A,P[A]}]

Equilateral

f[x_,y_]:=Sqrt[3] (1+3 x)>3 y&&Sqrt[3] +6 y>0&&Sqrt[3] >3 (Sqrt[3] x+y)

minx=-.5;
maxx=.5;
miny=-.3;
maxy=.6;

Length[pts=Partition[Select[Table[{RandomReal[{minx,maxx}],RandomReal[{miny,maxy}]},{10^5}],f@@#&],3]]//Timing

Graphics[Transpose[{{Red,Green,Blue},Point/@Transpose[Take[pts,10^3]]}]]

areas=TriangleArea/@pts;//Timing

Through[{Min,Mean,Variance,Max}[areas]]

Histogram[areas,AxesLabel->{A,P[A]}]

Right

n=5000;
{Plus@@Abs[TriangleArea/@Table[RandomTrianglePoints[Rest[rt],3],{n}]]/Abs[TriangleArea[rt]]/n,1/12.}//Timing

This Is Affine

t=Triangle[Table[Random[],{3},{2}]];
Length[pts=Select[Table[Random[],{10^6},{2}],TriangleInteriorQ[#,t]&]]//Timing

tris=Partition[pts,3];

Show[Graphics[
{
{Red,Triangle/@pairs},
t
},AspectRatio->Automatic]]

{Length[tris],Mean[Area[Evaluate[Triangle[#]]]&/@tris],Area[Evaluate[t]]}

l={
{Floor[14996/3],0.006672284988305263`,0.07863009404546599`},
{1709,0.004272266159857728`,0.050540650440876064`},
{14433,0.0036544337994560217`,0.043277463058226474`}
};

#[[2]]/#[[3]]&/@l

N[1/12,10]

Numerical

TriangleArea[{{x1,y1},{x2,y2},{x3,y3}}]

8NIntegrate[ Abs[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3],{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3},Method->QuasiMonteCarlo,PrecisionGoal->3]

8NIntegrate[(Sqrt[(((-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3))^2)]),{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3},Method->QuasiMonteCarlo,PrecisionGoal->3]

1/12.

Integrate[ Abs[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3],{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3},

Exact Form from Philip

CDF

(Subscript[G, 3])[u_]:=15(u^2)-24(u^(5/2))+10(u^3)
(Subscript[H, 3,1])[u_]:=4u-3(u^2)+2(u^2)Log[u]
(Subscript[H, 3,2])[u_]:=Piecewise[{{(1/3)Log[((1-3u+(1-u)(Sqrt[1-4u]))/(2(u^(3/2))))](((1-4u))^(3/2))+(1/2)(1-6u)Log[u]+u,0≤u≤1/4},
{(1/3)(ArcCos[((1-3u)/(2(u^(3/2))))]-π)(((4u-1))^(3/2))+(1/2)(1-6u)Log[u]+u,1/4<u≤1}}]
(Subscript[H, 3])[u_]:=(3/5)(Subscript[H, 3,1])[u]+(2/5)(Subscript[H, 3,2])[u]

(i=(Subscript[G, 3])[x]-Integrate[(Subscript[H, 3])[u]D[(Subscript[G, 3])[(x/u)],u],{u,x,1},Assumptions->0≤x≤1])//Timing

d[x_]:=Piecewise[{
{8(2(x^3)+3(x^2))(3Log[((1+(Sqrt[1-4x]))/2)]Log[((1+(Sqrt[1-4x]))/(2x))]-((π^2)/3))+(2/5)(324(x^2)+28x-1)((1/2)Log[x]-Log[((1+(Sqrt[1-4x]))/2)])(Sqrt[1-4x])+12(x^3)((Log[x])^2)-(54(x^2)+6x-(1/5))Log[x]-(57/5)(x^2)+(62/5)x,0≤x≤1/4},
{8(2(x^3)+3(x^2))((2π-3ArcCos[(1/(2(Sqrt[x])))])ArcCos[(1/(2(Sqrt[x])))]-((π^2)/3))+(2/5)(324(x^2)+28x-1)(ArcCos[(1/(2(Sqrt[x])))]-(π/3))(Sqrt[4x-1])-18(x^2)((Log[x])^2)-(54(x^2)+6x-(1/5))Log[x]-(57/5)(x^2)+(62/5)x,1/4<x≤1}}]

Limit[d[x],x->0,Direction->-1]

<<Utilities`Simplify`

Limit[d[x],x->1/4,Direction->#]&/@{-1,1}//LogExpand

Limit[d[x],x->1,Direction->1]

Plot[d[x],{x,0,1},PlotStyle->Red,PlotRange->All,Exclusions->None]

PDF

FullSimplify[d'[x],0<x<1]

<<Utilities`Simplify`

FullSimplify[d'[x],0<x<1]//LogExpand

p[x_]:=(Piecewise[{{-(1/(1+(Sqrt[1-4 x])-4 x))4 (-3 (1+(Sqrt[1-4 x]))-3 Log[2]+x (4 (π^2) (1+(Sqrt[1-4 x])-4 x) (1+x)+3 (5+(Sqrt[1-4 x])-22 Log[2]+4 x (-1+26 Log[2])))+(3+6 (11-52 x) x) Log[1+(Sqrt[1-4 x])]+3 x Log[x] (-1+(Sqrt[1-4 x])+(4+52 (Sqrt[1-4 x])) x+3 x (-1-(Sqrt[1-4 x])+4 x) Log[x])-3 (Log[2]-Log[1+(Sqrt[1-4 x])]) ((Sqrt[1-4 x]) (1+2 (11-52 x) x)+12 (1+(Sqrt[1-4 x])-4 x) x (1+x) (Log[2]-Log[1+(Sqrt[1-4 x])]+Log[x]))), 4 x<1}, {2 (6-2 (π^2) (x^2)+π (Sqrt[-1+4 x])+26 π x (Sqrt[-1+4 x])-6 (-4 π x (1+x)+(Sqrt[-1+4 x]) (1+26 x)) ArcCsc[2 (Sqrt[x])]-72 x (1+x) ((ArcCsc[2 (Sqrt[x])])^2)-3 Log[x]-2 x (3+(π^2)+9 Log[x] (4+Log[x]))), 4 x>1}}])

NIntegrate[x p[x],{x,0,1}]

FullSimplify[p[x],0<x<1]

Simplify[(expr/.{(Sqrt[1-4 x])-4 x->t,-((Sqrt[1-4 x])-4 x)->-t})/.(Sqrt[1-4 x]) ->s-1,0<x<1]//TraditionalForm

Plot[p[x],{x,0,1},PlotStyle->Red,PlotRange->All,Exclusions->None]

Integrate[p[x],{x,0,1}]//FullSimplify//Timing

Plots

GraphicsGrid[{Plot[#1[x],{x,0,1},PlotStyle->Red,PlotRange->All,Exclusions->None,AxesLabel->#2[A]]&@@@{{p,PDF},{d,CDF}}},ImageSize->500,Spacings->-.1]

Raw Moments

Integrate[1,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}]

p[x]

(muraw=Table[Integrate[x^n p[x],{x,0,1}],{n,0,25}]//FullSimplify)//Timing

(i=Integrate[x^n p[x],{x,0,1},Assumptions->n≥0])//Timing

(i=Integrate[x^n p[x],{x,0,1}])//Timing

(i=Integrate[x^n p[x],x])//Timing

(i1=Integrate[#,{x,0,1},Assumptions->n≥0]&/@Expand[Simplify[x^n p[x],0<x<1/4]])//Timing

(i2=Integrate[#,{x,0,1},Assumptions->n≥0]&/@Expand[Simplify[x^n p[x],1/4<x<1]])//Timing

(ipart1=Integrate[#,x]&/@Expand[Simplify[x^n p[x],0<x<1/4]])//Timing

Simplify[ipart1,0≤x≤1/4&&n∈Integers&&n≥0]

res1=%;

res1/.n->0

i11=Limit[ipart1,x->0,Direction->-1]

i12=Limit[ipart1,x->1/4,Direction->1]

(ipart2=Integrate[#,x]&/@Expand[Simplify[x^n p[x],1/4<x<1]])//Timing

i21=Limit[ipart1,x->1/4,Direction->-1]

i22=Limit[ipart1,x->1,Direction->1]

1

Simplify`PiecewiseIntegrate;
PiecewiseDump`$ShowIntegrals=True;
i=Integrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)],{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}];

Length[i]

Timing[8i/.HoldForm[Integrate[x__]]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo,MaxPoints->10^6]]

v4.2 with Last one done as Indefinite

Most[l_List]:=Drop[l,-1]

l={};
Do[
Print[{n,(v=i[[n]]/.integrate:>IntegrateWithAssumptions2)//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
]

(Subscript[I, 14]) and (Subscript[I, 47]) have a spurious imaginary part

Position[l,_?(Im[#]!=0&),{1}]//Flatten

Length[l]

8FullSimplify[Plus@@Delete[l,List/@{14,47}]+(-(277/5760)+((5 Log[2])/72))+(-(277/5760)+((5 Log[2])/72))]

v4.2

Most[l_List]:=Drop[l,-1]

l={};
Do[
Print[{n,(v=i[[n]]/.integrate:>IntegrateWithAssumptions)//Timing,AppendTo[l,v];
If[FreeQ[v,Integrate],N[v],$Failed]}],
{n,Length[i]}
]

Simplify[8Plus@@l]

v5

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.integrate:>IntegrateWithAssumptions,120])//Timing,AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,Length[i]}
]

v6 with Last one done as Indefinite

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.integrate:>IntegrateWithAssumptions2,120])//Timing,AppendTo[l,v];
If[FreeQ[v,($Aborted|Integrate)],N[v],$Failed]}],
{n,Length[i]}
]

v6

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.integrate:>IntegrateWithAssumptions,60])//Timing,AppendTo[l,v];
If[FreeQ[v,($Aborted|Integrate)],N[v],$Failed]}],
{n,Length[i]}
]

ints={(1/3456),(1/1728),((-589+850 Log[2])/5760),((-7277+10500 Log[2])/17280),-(131/1728)+((7 Log[2])/64),(37/640)-((Log[2])/12),(1/1152),0,0,0,0,0,-(131/1728)+((7 Log[2])/64),0,0,0,0,-(145949/5760)+((329 Log[2])/9),-(1132273/30240)+((50 (π^2))/9)-((1012 Log[2])/21)+((100 ((Log[2])^2))/3),(2411287/30240)-((100 (π^2))/9)+((1877 Log[2])/21)-((200 ((Log[2])^2))/3),0,0,0,0,((80497-12040 (π^2)+105376 Log[2]-72240 ((Log[2])^2))/40320),((-7817-840 π (I+86 π)+288 (4654-1505 Log[2]) Log[2])/120960),(((1219075/12)-16555 (π^2)+158012 Log[2]-99330 ((Log[2])^2))/5040),-(199413/4480)+((43 (π^2))/9)-((1492 Log[2])/63)+((86 ((Log[2])^2))/3),(72977/2880)-((329 Log[2])/9),(4529183/120960)-((50 (π^2))/9)+((1012 Log[2])/21)-((100 ((Log[2])^2))/3),-(4822567/60480)+((100 (π^2))/9)-((1877 Log[2])/21)+((200 ((Log[2])^2))/3),(1/3456),(1/1728),-(131/1728)+((7 Log[2])/64),((-7277+10500 Log[2])/17280),((-589+850 Log[2])/5760),(37/640)-((Log[2])/12),(1/1152),0,0,0,0,0,0,0,0,0,0,-(145949/5760)+((329 Log[2])/9),-(1132273/30240)+((50 (π^2))/9)-((1012 Log[2])/21)+((100 ((Log[2])^2))/3),(2411287/30240)-((100 (π^2))/9)+((1877 Log[2])/21)-((200 ((Log[2])^2))/3),0,0,0,0,(((1219075/12)-16555 (π^2)+158012 Log[2]-99330 ((Log[2])^2))/5040),((-7817-840 π (I+86 π)+288 (4654-1505 Log[2]) Log[2])/120960),((80497-12040 (π^2)+105376 Log[2]-72240 ((Log[2])^2))/40320),-(199413/4480)+((43 (π^2))/9)-((1492 Log[2])/63)+((86 ((Log[2])^2))/3),(72977/2880)-((329 Log[2])/9),(4529183/120960)-((50 (π^2))/9)+((1012 Log[2])/21)-((100 ((Log[2])^2))/3),-(4822567/60480)+((100 (π^2))/9)-((1877 Log[2])/21)+((200 ((Log[2])^2))/3)};

Length[ints]

zeros=Flatten[Position[ints,0]]

zeros={8,9,10,11,12,14,15,16,17,21,22,23,24,39,40,41,42,43,44,45,46,47,48,52,53,54,55};

v4.2 timeconstrained 1 minute

sym={};
num={};
Do[
Print[
{
n,
Timing[Last@AppendTo[sym,TimeConstrained[i[[n]]/.integrate:>Integrate,120,0]]],
N[Last[sym]],
Timing[Last@AppendTo[num,TimeConstrained[i[[n]]/.integrate[x__]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo],60,0]]]
}
],
{n,Length[i]}
]

v6 timeconstrained 1 minute

sym={};
num={};
Do[
Print[
{
n,
Timing[Last@AppendTo[sym,TimeConstrained[i[[n]]/.HoldForm[Integrate[x__]]:>IntegrateWithAssumptions[x],60,0]]],
N[Last[sym]],
Timing[Last@AppendTo[num,TimeConstrained[i[[n]]/.HoldForm[Integrate[x__]]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo,WorkingPrecision->20],60,0]]]
}
],
{n,Length[i]}
]

sym={};
num={};
Do[
Print[
{
zeros[[n]],
Timing[Last@AppendTo[sym,TimeConstrained[i[[zeros[[n]]]]/.integrate:>Integrate,10 60,0]]],
N[Last[sym]],
Timing[Last@AppendTo[num,TimeConstrained[i[[zeros[[n]]]]/.integrate[x__]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo,WorkingPrecision->20],5 60,0]]]
}
],
{n,Length[zeros]}
]

ints={(1/3456),(1/1728),((-589+850 Log[2])/5760),((-7277+10500 Log[2])/17280),-(131/1728)+((7 Log[2])/64),(37/640)-((Log[2])/12),(1/1152),0,0,0,0,0,-(131/1728)+((7 Log[2])/64),0,0,-(2557/8640)+((41 Log[2])/96),(1/576) (-131+189 Log[2]),-(145949/5760)+((329 Log[2])/9),-(1132273/30240)+((50 (π^2))/9)-((1012 Log[2])/21)+((100 ((Log[2])^2))/3),(2411287/30240)-((100 (π^2))/9)+((1877 Log[2])/21)-((200 ((Log[2])^2))/3),((-119221+18060 (π^2)-160263 Log[2]+108360 ((Log[2])^2))/60480),((16489+24080 (π^2)-466804 Log[2]+144480 ((Log[2])^2))/40320),((-(2422571/6)+66220 (π^2)-635791 Log[2]+397320 ((Log[2])^2))/20160),(5389463/120960)-((43 (π^2))/9)+((29761 Log[2])/1260)-((86 ((Log[2])^2))/3),((80497-12040 (π^2)+105376 Log[2]-72240 ((Log[2])^2))/40320),((-7817-840 π (I+86 π)+288 (4654-1505 Log[2]) Log[2])/120960),(((1219075/12)-16555 (π^2)+158012 Log[2]-99330 ((Log[2])^2))/5040),-(199413/4480)+((43 (π^2))/9)-((1492 Log[2])/63)+((86 ((Log[2])^2))/3),(72977/2880)-((329 Log[2])/9),(4529183/120960)-((50 (π^2))/9)+((1012 Log[2])/21)-((100 ((Log[2])^2))/3),-(4822567/60480)+((100 (π^2))/9)-((1877 Log[2])/21)+((200 ((Log[2])^2))/3),(1/3456),(1/1728),-(131/1728)+((7 Log[2])/64),((-7277+10500 Log[2])/17280),((-589+850 Log[2])/5760),(37/640)-((Log[2])/12),(1/1152),0,0,0,0,0,(1/576) (-131+189 Log[2]),-(2557/8640)+((41 Log[2])/96),(1/576) (61-88 Log[2]),0,-(131/1728)+((7 Log[2])/64),-(145949/5760)+((329 Log[2])/9),-(1132273/30240)+((50 (π^2))/9)-((1012 Log[2])/21)+((100 ((Log[2])^2))/3),(2411287/30240)-((100 (π^2))/9)+((1877 Log[2])/21)-((200 ((Log[2])^2))/3),((-(2422571/6)+66220 (π^2)-635791 Log[2]+397320 ((Log[2])^2))/20160),((16489+24080 (π^2)-466804 Log[2]+144480 ((Log[2])^2))/40320),((-119221+18060 (π^2)-160263 Log[2]+108360 ((Log[2])^2))/60480),(5389463/120960)-((43 (π^2))/9)+((29761 Log[2])/1260)-((86 ((Log[2])^2))/3),(((1219075/12)-16555 (π^2)+158012 Log[2]-99330 ((Log[2])^2))/5040),((-7817-840 π (I+86 π)+288 (4654-1505 Log[2]) Log[2])/120960),((80497-12040 (π^2)+105376 Log[2]-72240 ((Log[2])^2))/40320),-(199413/4480)+((43 (π^2))/9)-((1492 Log[2])/63)+((86 ((Log[2])^2))/3),(72977/2880)-((329 Log[2])/9),(4529183/120960)-((50 (π^2))/9)+((1012 Log[2])/21)-((100 ((Log[2])^2))/3),-(4822567/60480)+((100 (π^2))/9)-((1877 Log[2])/21)+((200 ((Log[2])^2))/3)};

Length[ints]

zeros=Position[ints,0]//Flatten

zeros={8,9,10,11,12,14,15,39,40,41,42,43,47};

sym={};
num={};
Do[
Print[
{
zeros[[n]],
Timing[Last@AppendTo[sym,TimeConstrained[i[[zeros[[n]]]]/.integrate:>Integrate,20 60,0]]],
N[Last[sym]],
Timing[Last@AppendTo[num,TimeConstrained[i[[zeros[[n]]]]/.integrate[x__]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo,WorkingPrecision->20],5 60,0]]]
}
],
{n,Length[zeros]}
]

Position[zeros,12]

sym={};
num={};
Do[
Print[
{
zeros[[n]],
Timing[Last@AppendTo[sym,TimeConstrained[i[[zeros[[n]]]]/.integrate:>Integrate,20 60,0]]],
N[Last[sym]],
Timing[Last@AppendTo[num,TimeConstrained[i[[zeros[[n]]]]/.integrate[x__]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo,WorkingPrecision->20],5 60,0]]]
}
],
{n,6,Length[zeros]}
]

n=6 crashes

sym={};
num={};
Do[
Print[
{
zeros[[n]],
Timing[Last@AppendTo[sym,TimeConstrained[i[[zeros[[n]]]]/.integrate:>Integrate,20 60,0]]],
N[Last[sym]],
Timing[Last@AppendTo[num,TimeConstrained[i[[zeros[[n]]]]/.integrate[x__]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo,WorkingPrecision->20],5 60,0]]]
}
],
{n,7,Length[zeros]}
]

Integrals that Are Wrong Or Hang in v6

Unless noted, these are done in v4.2

8

v4.2

TimeConstrained[i[[8]]/.integrate:>Integrate,120]//Timing

9

v4.0

TimeConstrained[i[[9]]/.integrate:>Integrate,20 60]//Timing

v4.2

TimeConstrained[i[[9]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[9]]/.integrate:>Integrate,20 60]//Timing

By hand

i[[9]]

i[[9]]/.integrate[x__]:>NIntegrate[Evaluate[x],Method->QuasiMonteCarlo]//Timing

Simplify[i1=Integrate[x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3,{y1,x1,((-x1+x2+x1 x2-x2 x3)/(x2-x3))},{y2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3)),1},{y3,x3,1}]]

NIntegrate[(1/(12 (x1-x3) (x2-x3)))((x1-x2) (((-1+x3))^2) (6 (x3^2)-12 x2 (x3^2)+(x1^2) (5-12 x2+6 (x2^2)+2 x3-(x3^2))+(x2^2) (-1+2 x3+5 (x3^2))+2 x1 (-6 x3-6 (x2^2) x3+x2 (1+10 x3+(x3^2))))),{x1,0,1},{x2,0,x1},{x3,((2 x1-x2-x1 x2)/(1+x1-2 x2)),1},Method->QuasiMonteCarlo]

i2=Simplify[Integrate[i1,{x3,((2 x1-x2-x1 x2)/(1+x1-2 x2)),1},Assumptions->0<x2<x1<1]]

i2s=Simplify[%,0<x2<x1<1]

NIntegrate[i2s,{x1,0,1},{x2,0,x1}]

i3=Simplify[Integrate[i2s,x2]]

int3=Simplify[Limit[i3,x2->x1,Direction->1]-Limit[i3,x2->0,Direction->-1]]

NIntegrate[int3,{x1,0,1}]

i4=Simplify[Integrate[int3,x1]]

int4=Simplify[Limit[i4,x1->1,Direction->1]-Limit[i4,x1->0,Direction->-1]]

N[%]

10

v4.2

TimeConstrained[i[[10]]/.integrate:>Integrate,120]//Timing

11

v4.0

TimeConstrained[i[[11]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[11]]/.integrate:>Integrate,20 60]//Timing

Simplify[((I (-1474 I+2088 I Log[2]))/60480)]

v4.2

TimeConstrained[i[[11]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[11]]/.integrate:>Integrate,20 60]//Timing

By Hand

i[[11]]

NIntegrate[x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3,{x1,0,1},{x2,0,x1},{x3,((2 x1-x2-x1 x2)/(1+x1-2 x2)),1},{y1,((-x1+x2+x1 x2-x2 x3)/(x2-x3)),((-x1+x3+x1 x3-x2 x3)/(-x2+x3))},{y2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3)),1},{y3,x3,1},Method->QuasiMonteCarlo,MaxPoints->10^6]

i1=Simplify[Integrate[x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3,{y1,((-x1+x2+x1 x2-x2 x3)/(x2-x3)),((-x1+x3+x1 x3-x2 x3)/(-x2+x3))},{y2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3)),1},{y3,x3,1}]]

NIntegrate[i1,{x1,0,1},{x2,0,x1},{x3,((2 x1-x2-x1 x2)/(1+x1-2 x2)),1},Method->QuasiMonteCarlo]

i2=Simplify[Integrate[i1,{x3,((2 x1-x2-x1 x2)/(1+x1-2 x2)),1},Assumptions->0<x2<x1<1]]

NIntegrate[i2,{x1,0,1},{x2,0,x1}]

i3=Simplify[Integrate[i2,x2]]

int3=Simplify[Limit[i3,x2->x1,Direction->1]-Limit[i3,x2->0,Direction->-1]]

NIntegrate[int3,{x1,0,1}]

i4=Simplify[Integrate[int3,x1]]

int4=Simplify[Limit[i4,x1->1,Direction->1]-Limit[i4,x1->0,Direction->-1]]

N[%]

12

v4.0

TimeConstrained[i[[12]]/.integrate:>Integrate,20 60]//Timing

v4.2

TimeConstrained[i[[12]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[12]]/.integrate:>Integrate,20 60]//Timing

By Hand

i[[12]]

NIntegrate[x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3,{x1,0,1},{x2,0,x1},{x3,((2 x1-x2-x1 x2)/(1+x1-2 x2)),1},{y1,((-x1+x3+x1 x3-x2 x3)/(-x2+x3)),1},{y2,((x1-x2+x2 y1-x3 y1)/(x1-x3)),1},{y3,((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)),1},Method->QuasiMonteCarlo,MaxPoints->10^6]

i1=Simplify[Integrate[x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3,{y1,((-x1+x3+x1 x3-x2 x3)/(-x2+x3)),1},{y2,((x1-x2+x2 y1-x3 y1)/(x1-x3)),1},{y3,((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2)),1}]]

i2=Simplify[Integrate[i1,{x3,((2 x1-x2-x1 x2)/(1+x1-2 x2)),1},Assumptions->0<x2<x1<1]]

NIntegrate[i2,{x1,0,1},{x2,0,x1},Method->QuasiMonteCarlo]

i3=Simplify[Integrate[i2,x2]]

int3=Simplify[Limit[i3,x2->x1,Direction->1]-Limit[i3,x2->0,Direction->-1]]

NIntegrate[int3,{x1,0,1}]

i4=Simplify[Integrate[int3,x1]]

int4=Simplify[Limit[i4,x1->1,Direction->1]-Limit[i4,x1->0,Direction->-1]]

N[%]

14

v4.0

TimeConstrained[i[[14]]/.integrate:>Integrate,20 60]//Timing

v4.2

TimeConstrained[i[[14]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[14]]/.integrate:>Integrate,20 60]//Timing

v6

crashes IA-64

By Hand

i[[14]]

NIntegrate[x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3,{x1,0,1},{x2,x1,((2 x1)/(1+x1))},{x3,0,((2 x1-x2-x1 x2)/(1+x1-2 x2))},{y1,((-x1+x2+x1 x2-x2 x3)/(x2-x3)),((-x1+x3+x1 x3-x2 x3)/(-x2+x3))},{y2,x2,((x1-x2+x2 y1-x3 y1)/(x1-x3))},{y3,x3,1},Method->QuasiMonteCarlo,MaxPoints->10^6]

i1=Simplify[Integrate[x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3,{y1,((-x1+x2+x1 x2-x2 x3)/(x2-x3)),((-x1+x3+x1 x3-x2 x3)/(-x2+x3))},{y2,x2,((x1-x2+x2 y1-x3 y1)/(x1-x3))},{y3,x3,1}]]

i2=Simplify[Integrate[i1,{x3,0,((2 x1-x2-x1 x2)/(1+x1-2 x2))},Assumptions->0<x1<1&&x1<x2<((2 x1)/(1+x1))]]

NIntegrate[i2,{x1,0,1},{x2,x1,((2 x1)/(1+x1))},Method->QuasiMonteCarlo]

i3=Simplify[Integrate[i2,x2]]

int3=Simplify[Limit[i3,x2->((2 x1)/(1+x1)),Direction->1]-Limit[i3,x2->x1,Direction->-1]]

NIntegrate[int3,{x1,0,1}]

i4=Simplify[Integrate[int3,x1]]

int4=Simplify[Limit[i4,x1->1,Direction->1]-Limit[i4,x1->0,Direction->-1]]

N[%]

15

v4.0

TimeConstrained[i[[15]]/.integrate:>Integrate,20 60]//Timing

v4.2

TimeConstrained[i[[15]]/.integrate:>Integrate,120]//Timing

v6

False 0

26

v4.0

TimeConstrained[i[[26]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[26]]/.integrate:>Integrate,20 60]//Timing

v4.2

TimeConstrained[i[[26]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[26]]/.integrate:>Integrate,20 60]//Timing

v6

Spurious imaginary part

By Hand

i[[26]]

NIntegrate[x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3,{x1,0,1},{x2,((2 x1)/(1+x1)),1},{x3,0,x1},{y1,((-x1+x2+x1 x2-x2 x3)/(x2-x3)),1},{y2,x2,((x1-x2+x2 y1-x3 y1)/(x1-x3))},{y3,x3,1},Method->QuasiMonteCarlo,MaxPoints->10^6]

i1=Simplify[Integrate[x2 y1-x3 y1-x1 y2+x3 y2+x1 y3-x2 y3,{y1,((-x1+x2+x1 x2-x2 x3)/(x2-x3)),1},{y2,x2,((x1-x2+x2 y1-x3 y1)/(x1-x3))},{y3,x3,1}]]

i2=Simplify[Integrate[i1,{x3,0,x1},Assumptions->0<x1<1&&((2 x1)/(1+x1))<x2<1]]

NIntegrate[i2,{x1,0,1},{x2,((2 x1)/(1+x1)),1}]

i3=Simplify[Integrate[i2,x2]]

int3=Simplify[Limit[i3,x2->1,Direction->1]-Limit[i3,x2->((2 x1)/(1+x1)),Direction->-1]]

i4=Simplify[Integrate[int3,x1]]

int4=Simplify[Limit[i4,x1->1,Direction->1]-Limit[i4,x1->0,Direction->-1]]

N[%]

39

v4.0

TimeConstrained[i[[39]]/.integrate:>Integrate,20 60]//Timing

v4.2

TimeConstrained[i[[39]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[39]]/.integrate:>Integrate,20 60]//Timing

By Hand

i[[39]]

NIntegrate[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3,{x1,0,1},{x2,0,x1},{x3,((2 x1-x2-x1 x2)/(1+x1-2 x2)),1},{y1,x1,((-x1+x2+x1 x2-x2 x3)/(x2-x3))},{y2,x2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3))},{y3,x3,((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))},Method->QuasiMonteCarlo,MaxPoints->10^6,PrecisionGoal->2]//Timing

i1=Simplify[Integrate[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3,{y1,x1,((-x1+x2+x1 x2-x2 x3)/(x2-x3))},{y2,x2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3))},{y3,x3,((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))}]]

...wrong in v6...  So use v5

i1=Simplify[Integrate[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3,{y1,x1,((-x1+x2+x1 x2-x2 x3)/(x2-x3))},{y2,x2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3))},{y3,x3,((x2 y1-x3 y1-x1 y2+x3 y2)/(-x1+x2))}]]

i2=Simplify[Integrate[i1,{x3,((2 x1-x2-x1 x2)/(1+x1-2 x2)),1},Assumptions->0<x2<x1<1]]

NIntegrate[i2,{x1,0,1},{x2,0,x1}]

i3=Simplify[Integrate[i2,x2]]

int3=Simplify[Limit[i3,x2->x1,Direction->1]-Limit[i3,x2->0,Direction->-1]]

NIntegrate[int3,{x1,0,1}]

i4=Simplify[Integrate[int3,x1]]

int4=Simplify[Limit[i4,x1->1,Direction->1]-Limit[i4,x1->0,Direction->-1]]

N[%]

40

v4.0

TimeConstrained[i[[40]]/.integrate:>Integrate,20 60]//Timing

v4.2

TimeConstrained[i[[40]]/.integrate:>Integrate,120]//Timing

N[((I (-1474 I+2088 I Log[2]))/60480)]

Simplify[((I (-1474 I+2088 I Log[2]))/60480)]

41

v4.2

TimeConstrained[i[[41]]/.integrate:>Integrate,120]//Timing

v6.0

False 0 in v6

42

v4.2

TimeConstrained[i[[42]]/.integrate:>Integrate,120]//Timing

43

v4.2

TimeConstrained[i[[43]]/.integrate:>Integrate,120]//Timing

47

v4.0

TimeConstrained[i[[47]]/.integrate:>Integrate,20 60]//Timing

v4.2

TimeConstrained[i[[47]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[47]]/.integrate:>Integrate,20 60]//Timing

By Hand

i[[47]]

NIntegrate[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3,{x1,0,1},{x2,x1,((2 x1)/(1+x1))},{x3,0,((2 x1-x2-x1 x2)/(1+x1-2 x2))},{y1,((-x1+x2+x1 x2-x2 x3)/(x2-x3)),((-x1+x3+x1 x3-x2 x3)/(-x2+x3))},{y2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3)),1},{y3,x3,1},Method->QuasiMonteCarlo,MaxPoints->10^6]

i1=Simplify[Integrate[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3,{y1,((-x1+x2+x1 x2-x2 x3)/(x2-x3)),((-x1+x3+x1 x3-x2 x3)/(-x2+x3))},{y2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3)),1},{y3,x3,1}]]

i2=Simplify[Integrate[i1,{x3,0,((2 x1-x2-x1 x2)/(1+x1-2 x2))},Assumptions->0<x1<1&&x1<x2<((2 x1)/(1+x1))]]

NIntegrate[i2,{x1,0,1},{x2,x1,((2 x1)/(1+x1))},Method->QuasiMonteCarlo]

i3=Simplify[Integrate[i2,x2]]

int3=Simplify[Limit[i3,x2->((2 x1)/(1+x1)),Direction->1]-Limit[i3,x2->x1,Direction->-1]]

NIntegrate[int3,{x1,0,1}]

i4=Simplify[Integrate[int3,x1]]

int4=Simplify[Limit[i4,x1->1,Direction->1]-Limit[i4,x1->0,Direction->-1]]

N[%]

57

v4.0

TimeConstrained[i[[57]]/.integrate:>Integrate,20 60]//Timing

v4.2

TimeConstrained[i[[57]]/.integrate:>Integrate,5 60]//Timing

TimeConstrained[i[[57]]/.integrate:>Integrate,20 60]//Timing

v6.0

Spurious imaginary part

By Hand

i[[57]]

NIntegrate[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3,{x1,0,1},{x2,((2 x1)/(1+x1)),1},{x3,0,x1},{y1,x1,((-x1+x3+x1 x3-x2 x3)/(-x2+x3))},{y2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3)),1},{y3,x3,1},Method->QuasiMonteCarlo,MaxPoints->10^6]

i1=Simplify[Integrate[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3,{y1,x1,((-x1+x3+x1 x3-x2 x3)/(-x2+x3))},{y2,((x1 x3-x2 x3+x2 y1-x3 y1)/(x1-x3)),1},{y3,x3,1}]]

i2=Simplify[Integrate[i1,{x3,0,x1},Assumptions->0<x1<1&&((2 x1)/(1+x1))<x2<1]]

NIntegrate[i2,{x1,0,1},{x2,((2 x1)/(1+x1)),1}]

i3=Simplify[Integrate[i2,x2]]

int3=Simplify[Limit[i3,x2->1,Direction->1]-Limit[i3,x2->((2 x1)/(1+x1)),Direction->-1]]

NIntegrate[int3,{x1,0,1}]

i4=Simplify[Integrate[int3,x1]]

int4=Simplify[Limit[i4,x1->1,Direction->1]-Limit[i4,x1->0,Direction->-1]]

N[%]

Pieces

pieces={{1,(1/3456)},{2,(1/1728)},{3,((-589+850 Log[2])/5760)},{4,((-7277+10500 Log[2])/17280)},{5,-(131/1728)+((7 Log[2])/64)},{6,(37/640)-((Log[2])/12)},{7,(1/1152)},{8,(((3049/8)-((2199 Log[2])/4))/5040)},{9,((168287-243960 Log[2]+221 Log[64])/423360)},{10,((394-568 Log[2])/20160)},{11,((737-1044 Log[2])/30240)},{12,((3049-4398 Log[2])/120960)},{13,-(131/1728)+((7 Log[2])/64)},{14,-(277/5760)+((5 Log[2])/72)},{15,(1/576) (61-88 Log[2])},{16,-(2557/8640)+((41 Log[2])/96)},{17,(1/576) (-131+189 Log[2])},{18,-(145949/5760)+((329 Log[2])/9)},{19,-(1132273/30240)+((50 (π^2))/9)-((1012 Log[2])/21)+((100 ((Log[2])^2))/3)},{20,(2411287/30240)-((100 (π^2))/9)+((1877 Log[2])/21)-((200 ((Log[2])^2))/3)},{21,((-119221+18060 (π^2)-160263 Log[2]+108360 ((Log[2])^2))/60480)},{22,((16489+24080 (π^2)-466804 Log[2]+144480 ((Log[2])^2))/40320)},{23,((-(2422571/6)+66220 (π^2)-635791 Log[2]+397320 ((Log[2])^2))/20160)},{24,(5389463/120960)-((43 (π^2))/9)+((29761 Log[2])/1260)-((86 ((Log[2])^2))/3)},{25,((80497-12040 (π^2)+105376 Log[2]-72240 ((Log[2])^2))/40320)},{26,-(7817/120960)-((43 (π^2))/72)+((2327 Log[2])/210)-((43 ((Log[2])^2))/12)},{27,(((1219075/12)-16555 (π^2)+158012 Log[2]-99330 ((Log[2])^2))/5040)},{28,-(199413/4480)+((43 (π^2))/9)-((1492 Log[2])/63)+((86 ((Log[2])^2))/3)},{29,(72977/2880)-((329 Log[2])/9)},{30,(4529183/120960)-((50 (π^2))/9)+((1012 Log[2])/21)-((100 ((Log[2])^2))/3)},{31,-(4822567/60480)+((100 (π^2))/9)-((1877 Log[2])/21)+((200 ((Log[2])^2))/3)},{32,(1/3456)},{33,(1/1728)},{34,-(131/1728)+((7 Log[2])/64)},{35,((-7277+10500 Log[2])/17280)},{36,((-589+850 Log[2])/5760)},{37,(37/640)-((Log[2])/12)},{38,(1/1152)},{39,((3049-4398 Log[2])/120960)},{40,((737-1044 Log[2])/30240)},{41,((394-568 Log[2])/20160)},{42,(((24041/40)-((17331 Log[2])/20))/1512)},{43,(((3049/8)-((2199 Log[2])/4))/5040)},{44,(1/576) (-131+189 Log[2])},{45,-(2557/8640)+((41 Log[2])/96)},{46,(1/576) (61-88 Log[2])},{47,-(277/5760)+((5 Log[2])/72)},{48,-(131/1728)+((7 Log[2])/64)},{49,-(145949/5760)+((329 Log[2])/9)},{50,-(1132273/30240)+((50 (π^2))/9)-((1012 Log[2])/21)+((100 ((Log[2])^2))/3)},{51,(2411287/30240)-((100 (π^2))/9)+((1877 Log[2])/21)-((200 ((Log[2])^2))/3)},{52,((-(2422571/6)+66220 (π^2)-635791 Log[2]+397320 ((Log[2])^2))/20160)},{53,((16489+24080 (π^2)-466804 Log[2]+144480 ((Log[2])^2))/40320)},{54,((-119221+18060 (π^2)-160263 Log[2]+108360 ((Log[2])^2))/60480)},{55,(5389463/120960)-((43 (π^2))/9)+((29761 Log[2])/1260)-((86 ((Log[2])^2))/3)},{56,(((1219075/12)-16555 (π^2)+158012 Log[2]-99330 ((Log[2])^2))/5040)},{57,-(7817/120960)-((43 (π^2))/72)+((2327 Log[2])/210)-((43 ((Log[2])^2))/12)},{58,((80497-12040 (π^2)+105376 Log[2]-72240 ((Log[2])^2))/40320)},{59,-(199413/4480)+((43 (π^2))/9)-((1492 Log[2])/63)+((86 ((Log[2])^2))/3)},{60,(72977/2880)-((329 Log[2])/9)},{61,(4529183/120960)-((50 (π^2))/9)+((1012 Log[2])/21)-((100 ((Log[2])^2))/3)},{62,-(4822567/60480)+((100 (π^2))/9)-((1877 Log[2])/21)+((200 ((Log[2])^2))/3)}};

Uncomputed

Position[Last/@pieces,_?(#===0||Im[#]!=0&),{1}]//Flatten

All Pieces Combines

8Simplify[Plus@@Last/@pieces]

Number of Different Pieces

diff=Sort[Union[Expand/@FullSimplify/@Last/@pieces,SameTest->(Equal[N[#1,20],N[#2,20]]&)],Less]

Length[%]

{N[#],#}&/@diff//ColumnForm

2

(2^2)Integrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^2,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}]//Timing

N[1/144]

3

(2^3)Integrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)]^3,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}]//Timing

v4.2 CAD

<<Calculus`Integration`

i=Calculus`Integration`Private`DebugIntegrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)]^3,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}];

...hangs...

i=HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,0,y1/(1-x1+y1)},{y2,0,(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,0,y1/(1-x1+y1)},{y2,(x2*y1)/x1,x2},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,0,y1/(1-x1+y1)},{y2,(x2*y1)/x1,x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,0,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,0,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(x2*y1)/x1,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(x2*y1)/x1,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),x2},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(x2*y1)/x1,(-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(x2*y1)/x1,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,0,(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,(x2*y1)/x1,(-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,(x2*y1)/x1,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,0,(-x1+y1)/(-1+y1)},{y2,0,(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,0,(-x1+y1)/(-1+y1)},{y2,(x2*y1)/x1,x2},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,0,(-x1+y1)/(-1+y1)},{y2,(x2*y1)/x1,x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,0,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,0,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(x2*y1)/x1,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(x2*y1)/x1,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),x2},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,(x2*y1)/x1,x2},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,(x2*y1)/x1,x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,(x1-x2-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(x2*y1)/x1,(-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(x2*y1)/x1,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[-(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,0,y1/(1-x1+y1)},{y2,0,(x2*y1)/x1},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,0,y1/(1-x1+y1)},{y2,0,(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,0,y1/(1-x1+y1)},{y2,(x2*y1)/x1,x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,0,(-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,0,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(x2*y1)/x1,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,x1,1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(x2*y1)/x1,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,(-x1+y1)/(-1+y1),x1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,0,(x2*y1)/x1},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,0,(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,(x2*y1)/x1,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,0,x1/2},{x2,y1/(1-x1+y1),(-x1+y1)/(-1+y1)},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,0,(-x1+y1)/(-1+y1)},{y2,0,(x2*y1)/x1},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,0,(-x1+y1)/(-1+y1)},{y2,0,(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,0,(-x1+y1)/(-1+y1)},{y2,(x2*y1)/x1,x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,0,(-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,0,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(x2*y1)/x1,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,0,(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,0,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2)},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,x1,1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,(x2*y1)/x1,x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,(x1-x2-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,(-x1+y1)/(-1+y1),y1/(1-x1+y1)},{y2,(x1-x2-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,0,(x1-x2-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(y1-y2),(x2*y1-x1*y2)/(-x1+x2+y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(x2*y1)/x1,(-y1+x2*y1)/(-1+x1)},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(-y1+x2*y1)/(-1+x1),x2},{x3,(x2*y1-x1*y2)/(-x1+x2+y1-y2),(x2*y1-x1*y2)/(y1-y2)},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,0,(x2*y1-x1*y2)/(y1-y2)},{y3,0,x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]]+HoldForm[Integrate[(x2*y1-x3*y1-x1*y2+x3*y2+x1*y3-x2*y3)^3,{x1,0,1},{y1,x1/2,x1},{x2,y1/(1-x1+y1),x1},{y2,(x1-x2-y1+x2*y1)/(-1+x1),(x2*y1)/x1},{x3,(x2*y1-x1*y2)/(y1-y2),1},{y3,(x2*y1-x3*y1-x1*y2+x3*y2)/(-x1+x2),x3},Assumptions->True,GenerateConditions->Automatic,PrincipalValue->False]];

v4 IntegrateWithAssumptions

Most[l_]:=Drop[l,-1]

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.HoldForm[Integrate[x__,opts___?OptionQ]]:>IntegrateWithAssumptions[x],600])//Timing,AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,Length[i]}
]

Length[i]

v5.1 IntegrateWithAssumptions

Simplify`PiecewiseIntegrate;
PiecewiseDump`$ShowIntegrals=True;
i=Integrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)]^3,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}];

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.HoldForm[Integrate[x__,opts___?OptionQ]]:>IntegrateWithAssumptions[x],600])//Timing,AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,Length[i]}
]

v5.1 IntegrateWithAssumptions2

Simplify`PiecewiseIntegrate;
PiecewiseDump`$ShowIntegrals=True;
i=Integrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)]^3,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}];

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.HoldForm[Integrate[x__,opts___?OptionQ]]:>IntegrateWithAssumptions2[x],600])//Timing,AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,Length[i]}
]

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.HoldForm[Integrate[x__,opts___?OptionQ]]:>IntegrateWithAssumptions2[x],600])//Timing,AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,30,Length[i]}
]

By Hand

l1={$Aborted,$Aborted,$Aborted,(7/4608000),((-41+60 Log[2])/432000),((1049197-1136640 Log[2])/12773376000),$Aborted,((437557-631260 Log[2])/9072000),((2039929-2942080 Log[2])/53222400),$Aborted,((-51383+74130 Log[2])/216000),$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,((91+7680 I π)/13824000),((43-60 Log[2])/432000),((-457837+1136640 Log[2])/12773376000),$Aborted,((-5209+7515 Log[2])/108000),((-30589211+44131200 Log[2])/798336000),$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,((43-60 Log[2])/432000),$Aborted,((-457837+1136640 Log[2])/12773376000),$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,(8100866766589/67060224000)-((65629 (π^2))/3780)+((2534786161 Log[2])/17463600)-((65629 ((Log[2])^2))/630),$Aborted,$Aborted,$Aborted,$Aborted,((2039929-2942080 Log[2])/53222400),$Aborted,((-41+60 Log[2])/432000),$Aborted,((1049197-1136640 Log[2])/12773376000),$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,-(564277/13824000)+((53 Log[2])/900),((-51383+74130 Log[2])/216000),$Aborted,$Aborted,$Aborted};

l2={$Aborted,$Aborted,$Aborted,(7/4608000),((-41+60 Log[2])/432000),$Aborted,$Aborted,((1049197-1136640 Log[2])/12773376000),$Aborted,$Aborted,((-51383+74130 Log[2])/216000),$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,0,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,0,$Aborted,$Aborted,(91/13824000),((43-60 Log[2])/432000),$Aborted,$Aborted,((-457837+1136640 Log[2])/12773376000),$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,((-457837+1136640 Log[2])/12773376000),$Aborted,$Aborted,((43-60 Log[2])/432000),$Aborted,$Aborted,((-39062143+56354760 Log[2])/18144000),((-15194535899+1412028160 (π^2)+4740975680 Log[2]-4220636160 ((Log[2])^2))/851558400),$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,0,$Aborted,$Aborted,((1049197-1136640 Log[2])/12773376000),$Aborted,$Aborted,((-41+60 Log[2])/432000),$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted,$Aborted};

p1=Position[l1,$Aborted]//Flatten

p2=Position[l2,_?(#==$Aborted||#==0&)]//Flatten

Complement[p1,p2]

Complement[p2,p1]

l2[[Complement[p1,p2]]]

Position[Transpose[{l1,l2}],_?(Subtract@@#≠0&),1]

i[[#]]&/@{8,26,30,72}/.HoldForm[Integrate[x__,opts___?OptionQ]]:>NIntegrate[x,Method->QuasiMonteCarlo]

{l1[[#]],l2[[#]]}&/@{8,26,30,72}

N[%]

l={{1,$Aborted},{2,$Aborted},{3,$Aborted},{4,(7/4608000)},{5,((-41+60 Log[2])/432000)},{6,((1049197-1136640 Log[2])/12773376000)},{7,$Aborted},{8,((437557-631260 Log[2])/9072000)},{9,((2039929-2942080 Log[2])/53222400)},{10,$Aborted},{11,((-51383+74130 Log[2])/216000)},{12,$Aborted},{13,$Aborted},{14,$Aborted},{15,$Aborted},{16,$Aborted},{17,-(12932377571/2554675200)+((299 (π^2))/360)-((21873491 Log[2])/2661120)+((16 ((Log[2])^2))/3)},{18,$Aborted},{19,$Aborted},{20,$Aborted},{21,$Aborted},{22,$Aborted},{23,$Aborted},{24,$Aborted},{25,$Aborted},{26,(91/13824000)},{27,((43-60 Log[2])/432000)},{28,((-457837+1136640 Log[2])/12773376000)},{29,$Aborted},{30,((-5209+7515 Log[2])/108000)},{31,((-30589211+44131200 Log[2])/798336000)},{32,$Aborted},{33,$Aborted},{34,$Aborted},{35,$Aborted},{36,$Aborted},{37,$Aborted},{38,$Aborted},{39,$Aborted},{40,$Aborted},{41,$Aborted},{42,$Aborted},{43,$Aborted},{44,$Aborted},{45,$Aborted},{46,$Aborted},{47,$Aborted},{48,$Aborted},{49,$Aborted},{50,((-457837+1136640 Log[2])/12773376000)},{51,$Aborted},{52,((43-60 Log[2])/432000)},{53,((43-60 Log[2])/432000)},{54,((-457837+1136640 Log[2])/12773376000)},{55,$Aborted},{56,,((-39062143+56354760 Log[2])/18144000)},{57,((-15194535899+1412028160 (π^2)+4740975680 Log[2]-4220636160 ((Log[2])^2))/851558400)},{58,$Aborted},{59,$Aborted},{60,$Aborted},{61,$Aborted},{62,$Aborted},{63,$Aborted},{64,$Aborted},{65,$Aborted},{66,$Aborted},{67,(8100866766589/67060224000)-((65629 (π^2))/3780)+((2534786161 Log[2])/17463600)-((65629 ((Log[2])^2))/630)},{68,$Aborted},{69,$Aborted},{70,$Aborted},{71,$Aborted},{72,((2039929-2942080 Log[2])/53222400)},{73,$Aborted},{74,((-41+60 Log[2])/432000)},{75,((-41+60 Log[2])/432000)},{76,((1049197-1136640 Log[2])/12773376000)},{77,$Aborted},{78,$Aborted},{79,$Aborted},{80,$Aborted},{81,$Aborted},{82,$Aborted},{83,$Aborted},{84,-(564277/13824000)+((53 Log[2])/900)},{85,((-51383+74130 Log[2])/216000)},{86,$Aborted},{87,$Aborted},{88,$Aborted}};

Position[Last/@l,_?(#==$Aborted||#==0&)]//Flatten

17

Integrate[(1/(2304000 (((-2+x1))^4) (x1^5)))((((-1+x1))^5) (-7741440 x1+3870720 (x1^2)-4354560 (x1^3)-5241600 (x1^4)-16392768 (x1^5)-12902400 (π^2) (x1^5)+88678656 (x1^6)+25804800 (π^2) (x1^6)-99910752 (x1^7)-19353600 (π^2) (x1^7)+44733704 (x1^8)+6451200 (π^2) (x1^8)-7386238 (x1^9)-806400 (π^2) (x1^9)+52752 (x1^10)+8165 (x1^11)+1580 (x1^12)+330 (x1^13)+60 (x1^14)+15482880 Log[2]-11612160 x1 Log[2]+10321920 (x1^2) Log[2]+8386560 (x1^3) Log[2]+29998080 (x1^4) Log[2]-108399360 (x1^5) Log[2]+98257920 (x1^6) Log[2]-36195840 (x1^7) Log[2]+4807680 (x1^8) Log[2]+3840 (x1^9) Log[2]+38707200 (x1^5) ((Log[2])^2)-77414400 (x1^6) ((Log[2])^2)+58060800 (x1^7) ((Log[2])^2)-19353600 (x1^8) ((Log[2])^2)+2419200 (x1^9) ((Log[2])^2)-5760 (((-2+x1))^4) (168+210 x1+280 (x1^2)+420 (x1^3)+840 (x1^4)+403 (x1^5)) Log[2-x1]-38707200 (x1^6) Log[x1]+67737600 (x1^7) Log[x1]-41932800 (x1^8) Log[x1]+10080000 (x1^9) Log[x1]-483840 (x1^10) Log[x1]-40320 (x1^11) Log[x1]-5760 (x1^12) Log[x1]-1080 (x1^13) Log[x1]-240 (x1^14) Log[x1]-60 (x1^15) Log[x1]-37079040 (x1^5) Log[-(x1/(-1+x1))]+74158080 (x1^6) Log[-(x1/(-1+x1))]-55618560 (x1^7) Log[-(x1/(-1+x1))]+18539520 (x1^8) Log[-(x1/(-1+x1))]-2317440 (x1^9) Log[-(x1/(-1+x1))]-38707200 (x1^5) ((Log[-(x1/(-1+x1))])^2)+77414400 (x1^6) ((Log[-(x1/(-1+x1))])^2)-58060800 (x1^7) ((Log[-(x1/(-1+x1))])^2)+19353600 (x1^8) ((Log[-(x1/(-1+x1))])^2)-2419200 (x1^9) ((Log[-(x1/(-1+x1))])^2)+37079040 (x1^5) Log[(((-2+x1) x1)/(-1+x1))]-35450880 (x1^6) Log[(((-2+x1) x1)/(-1+x1))]-12119040 (x1^7) Log[(((-2+x1) x1)/(-1+x1))]+23393280 (x1^8) Log[(((-2+x1) x1)/(-1+x1))]-7762560 (x1^9) Log[(((-2+x1) x1)/(-1+x1))]+483840 (x1^10) Log[(((-2+x1) x1)/(-1+x1))]+40320 (x1^11) Log[(((-2+x1) x1)/(-1+x1))]+5760 (x1^12) Log[(((-2+x1) x1)/(-1+x1))]+1080 (x1^13) Log[(((-2+x1) x1)/(-1+x1))]+240 (x1^14) Log[(((-2+x1) x1)/(-1+x1))]+60 (x1^15) Log[(((-2+x1) x1)/(-1+x1))]-77414400 (x1^5) Log[2] Log[(((-2+x1) x1)/(-1+x1))]+154828800 (x1^6) Log[2] Log[(((-2+x1) x1)/(-1+x1))]-116121600 (x1^7) Log[2] Log[(((-2+x1) x1)/(-1+x1))]+38707200 (x1^8) Log[2] Log[(((-2+x1) x1)/(-1+x1))]-4838400 (x1^9) Log[2] Log[(((-2+x1) x1)/(-1+x1))]+38707200 (x1^5) ((Log[(((-2+x1) x1)/(-1+x1))])^2)-77414400 (x1^6) ((Log[(((-2+x1) x1)/(-1+x1))])^2)+58060800 (x1^7) ((Log[(((-2+x1) x1)/(-1+x1))])^2)-19353600 (x1^8) ((Log[(((-2+x1) x1)/(-1+x1))])^2)+2419200 (x1^9) ((Log[(((-2+x1) x1)/(-1+x1))])^2)+4838400 (((-2+x1))^4) (x1^5) PolyLog[2,1-(x1/2)])),{x1,0,1}]//Simplify

23

Integrate[(1/(2304000 (((-2+x1))^4) (x1^5)))(-7741440 x1+38707200 (x1^2)-82736640 (x1^3)+95719680 (x1^4)-71832768 (x1^5)+161601216 (x1^6)-595934112 (x1^7)+1235020824 (x1^8)-1491222618 (x1^9)+1104820650 (x1^10)-501811769 (x1^11)+131442233 (x1^12)-16419826 (x1^13)+350402 (x1^14)+40814 (x1^15)-6490 (x1^16)+3940 (x1^17)-660 (x1^18)+120 (x1^19)-15482880 Log[2]+89026560 x1 Log[2]-223211520 (x1^2) Log[2]+314173440 (x1^3) Log[2]-284820480 (x1^4) Log[2]+351348480 (x1^5) Log[2]-920021760 (x1^6) Log[2]+1881169920 (x1^7) Log[2]-2396298240 (x1^8) Log[2]+1943182080 (x1^9) Log[2]-1011628800 (x1^10) Log[2]+328204800 (x1^11) Log[2]-60480000 (x1^12) Log[2]+4838400 (x1^13) Log[2]-38707200 (x1^5) ((Log[2])^2)+270950400 (x1^6) ((Log[2])^2)-832204800 (x1^7) ((Log[2])^2)+1470873600 (x1^8) ((Log[2])^2)-1647475200 (x1^9) ((Log[2])^2)+1212019200 (x1^10) ((Log[2])^2)-585446400 (x1^11) ((Log[2])^2)+179020800 (x1^12) ((Log[2])^2)-31449600 (x1^13) ((Log[2])^2)+2419200 (x1^14) ((Log[2])^2)-15482880 Log[1-x1]+89026560 x1 Log[1-x1]-223211520 (x1^2) Log[1-x1]+314173440 (x1^3) Log[1-x1]-284820480 (x1^4) Log[1-x1]+351348480 (x1^5) Log[1-x1]-881314560 (x1^6) Log[1-x1]+1619896320 (x1^7) Log[1-x1]-1628605440 (x1^8) Log[1-x1]+658990080 (x1^9) Log[1-x1]+329495040 (x1^10) Log[1-x1]-571696320 (x1^11) Log[1-x1]+322357440 (x1^12) Log[1-x1]-91936680 (x1^13) Log[1-x1]+12111960 (x1^14) Log[1-x1]-289320 (x1^15) Log[1-x1]-50280 (x1^16) Log[1-x1]+12360 (x1^17) Log[1-x1]-4920 (x1^18) Log[1-x1]+840 (x1^19) Log[1-x1]-120 (x1^20) Log[1-x1]-77414400 (x1^5) Log[2] Log[1-x1]+541900800 (x1^6) Log[2] Log[1-x1]-1664409600 (x1^7) Log[2] Log[1-x1]+2941747200 (x1^8) Log[2] Log[1-x1]-3294950400 (x1^9) Log[2] Log[1-x1]+2424038400 (x1^10) Log[2] Log[1-x1]-1170892800 (x1^11) Log[2] Log[1-x1]+358041600 (x1^12) Log[2] Log[1-x1]-62899200 (x1^13) Log[2] Log[1-x1]+4838400 (x1^14) Log[2] Log[1-x1]-38707200 (x1^5) ((Log[1-x1])^2)+270950400 (x1^6) ((Log[1-x1])^2)-832204800 (x1^7) ((Log[1-x1])^2)+1470873600 (x1^8) ((Log[1-x1])^2)-1647475200 (x1^9) ((Log[1-x1])^2)+1212019200 (x1^10) ((Log[1-x1])^2)-585446400 (x1^11) ((Log[1-x1])^2)+179020800 (x1^12) ((Log[1-x1])^2)-31449600 (x1^13) ((Log[1-x1])^2)+2419200 (x1^14) ((Log[1-x1])^2)+15482880 Log[2-x1]-89026560 x1 Log[2-x1]+223211520 (x1^2) Log[2-x1]-314173440 (x1^3) Log[2-x1]+284820480 (x1^4) Log[2-x1]-351348480 (x1^5) Log[2-x1]+881314560 (x1^6) Log[2-x1]-1619896320 (x1^7) Log[2-x1]+1628605440 (x1^8) Log[2-x1]-658990080 (x1^9) Log[2-x1]-329495040 (x1^10) Log[2-x1]+571696320 (x1^11) Log[2-x1]-322357440 (x1^12) Log[2-x1]+91936680 (x1^13) Log[2-x1]-12111960 (x1^14) Log[2-x1]+289320 (x1^15) Log[2-x1]+50280 (x1^16) Log[2-x1]-12360 (x1^17) Log[2-x1]+4920 (x1^18) Log[2-x1]-840 (x1^19) Log[2-x1]+120 (x1^20) Log[2-x1]+77414400 (x1^5) Log[2] Log[2-x1]-541900800 (x1^6) Log[2] Log[2-x1]+1664409600 (x1^7) Log[2] Log[2-x1]-2941747200 (x1^8) Log[2] Log[2-x1]+3294950400 (x1^9) Log[2] Log[2-x1]-2424038400 (x1^10) Log[2] Log[2-x1]+1170892800 (x1^11) Log[2] Log[2-x1]-358041600 (x1^12) Log[2] Log[2-x1]+62899200 (x1^13) Log[2] Log[2-x1]-4838400 (x1^14) Log[2] Log[2-x1]+77414400 (x1^5) Log[1-x1] Log[2-x1]-541900800 (x1^6) Log[1-x1] Log[2-x1]+1664409600 (x1^7) Log[1-x1] Log[2-x1]-2941747200 (x1^8) Log[1-x1] Log[2-x1]+3294950400 (x1^9) Log[1-x1] Log[2-x1]-2424038400 (x1^10) Log[1-x1] Log[2-x1]+1170892800 (x1^11) Log[1-x1] Log[2-x1]-358041600 (x1^12) Log[1-x1] Log[2-x1]+62899200 (x1^13) Log[1-x1] Log[2-x1]-4838400 (x1^14) Log[1-x1] Log[2-x1]-38707200 (x1^5) ((Log[2-x1])^2)+270950400 (x1^6) ((Log[2-x1])^2)-832204800 (x1^7) ((Log[2-x1])^2)+1470873600 (x1^8) ((Log[2-x1])^2)-1647475200 (x1^9) ((Log[2-x1])^2)+1212019200 (x1^10) ((Log[2-x1])^2)-585446400 (x1^11) ((Log[2-x1])^2)+179020800 (x1^12) ((Log[2-x1])^2)-31449600 (x1^13) ((Log[2-x1])^2)+2419200 (x1^14) ((Log[2-x1])^2)+77414400 (x1^5) Log[2] Log[x1]-541900800 (x1^6) Log[2] Log[x1]+1664409600 (x1^7) Log[2] Log[x1]-2941747200 (x1^8) Log[2] Log[x1]+3294950400 (x1^9) Log[2] Log[x1]-2424038400 (x1^10) Log[2] Log[x1]+1170892800 (x1^11) Log[2] Log[x1]-358041600 (x1^12) Log[2] Log[x1]+62899200 (x1^13) Log[2] Log[x1]-4838400 (x1^14) Log[2] Log[x1]+77414400 (x1^5) Log[1-x1] Log[x1]-541900800 (x1^6) Log[1-x1] Log[x1]+1664409600 (x1^7) Log[1-x1] Log[x1]-2941747200 (x1^8) Log[1-x1] Log[x1]+3294950400 (x1^9) Log[1-x1] Log[x1]-2424038400 (x1^10) Log[1-x1] Log[x1]+1170892800 (x1^11) Log[1-x1] Log[x1]-358041600 (x1^12) Log[1-x1] Log[x1]+62899200 (x1^13) Log[1-x1] Log[x1]-4838400 (x1^14) Log[1-x1] Log[x1]-77414400 (x1^5) Log[2-x1] Log[x1]+541900800 (x1^6) Log[2-x1] Log[x1]-1664409600 (x1^7) Log[2-x1] Log[x1]+2941747200 (x1^8) Log[2-x1] Log[x1]-3294950400 (x1^9) Log[2-x1] Log[x1]+2424038400 (x1^10) Log[2-x1] Log[x1]-1170892800 (x1^11) Log[2-x1] Log[x1]+358041600 (x1^12) Log[2-x1] Log[x1]-62899200 (x1^13) Log[2-x1] Log[x1]+4838400 (x1^14) Log[2-x1] Log[x1]-4838400 (((-2+x1))^4) (((-1+x1))^5) (x1^5) PolyLog[2,1-x1]+4838400 (((-2+x1))^4) (((-1+x1))^5) (x1^5) PolyLog[2,1-(x1/2)]),{x1,0,1}]//Simplify

69

Integrate[(1/(2304000 (((-2+x1))^4) (x1^5)))(7741440 x1-61931520 (x1^2)+224663040 (x1^3)-485856000 (x1^4)+702057408 (x1^5)-848067840 (x1^6)+1510183520 (x1^7)-3717370680 (x1^8)+7480001090 (x1^9)-10747873128 (x1^10)+11062112047 (x1^11)-8260538969 (x1^12)+4487573738 (x1^13)-1755366296 (x1^14)+481478792 (x1^15)-88032890 (x1^16)+9752400 (x1^17)-533700 (x1^18)+7560 (x1^19)+15482880 Log[2]-135475200 x1 Log[2]+541900800 (x1^2) Log[2]-1306368000 (x1^3) Log[2]+2125286400 (x1^4) Log[2]-2657237760 (x1^5) Log[2]+3528929280 (x1^6) Log[2]-6404616960 (x1^7) Log[2]+11801329920 (x1^8) Log[2]-17068275840 (x1^9) Log[2]+18301063680 (x1^10) Log[2]-14527946880 (x1^11) Log[2]+8556437760 (x1^12) Log[2]-3717390720 (x1^13) Log[2]+1170685440 (x1^14) Log[2]-258230400 (x1^15) Log[2]+37478400 (x1^16) Log[2]-3168000 (x1^17) Log[2]+115200 (x1^18) Log[2]+38707200 (x1^5) ((Log[2])^2)-387072000 (x1^6) ((Log[2])^2)+1774080000 (x1^7) ((Log[2])^2)-4935168000 (x1^8) ((Log[2])^2)+9299520000 (x1^9) ((Log[2])^2)-12542054400 (x1^10) ((Log[2])^2)+12468211200 (x1^11) ((Log[2])^2)-9266342400 (x1^12) ((Log[2])^2)+5163436800 (x1^13) ((Log[2])^2)-2141337600 (x1^14) ((Log[2])^2)+648403200 (x1^15) ((Log[2])^2)-138355200 (x1^16) ((Log[2])^2)+19526400 (x1^17) ((Log[2])^2)-1612800 (x1^18) ((Log[2])^2)+57600 (x1^19) ((Log[2])^2)+15482880 Log[1-x1]-135475200 x1 Log[1-x1]+541900800 (x1^2) Log[1-x1]-1306368000 (x1^3) Log[1-x1]+2125286400 (x1^4) Log[1-x1]-2657237760 (x1^5) Log[1-x1]+3490222080 (x1^6) Log[1-x1]-6027221760 (x1^7) Log[1-x1]+10120792320 (x1^8) Log[1-x1]-12545581440 (x1^9) Log[1-x1]+10099107840 (x1^10) Log[1-x1]-3950251200 (x1^11) Log[1-x1]-1417665600 (x1^12) Log[1-x1]+3241036200 (x1^13) Log[1-x1]-2419681200 (x1^14) Log[1-x1]+1095170400 (x1^15) Log[1-x1]-325363080 (x1^16) Log[1-x1]+62721960 (x1^17) Log[1-x1]-7297920 (x1^18) Log[1-x1]+429840 (x1^19) Log[1-x1]-7560 (x1^20) Log[1-x1]+77414400 (x1^5) Log[2] Log[1-x1]-774144000 (x1^6) Log[2] Log[1-x1]+3548160000 (x1^7) Log[2] Log[1-x1]-9870336000 (x1^8) Log[2] Log[1-x1]+18599040000 (x1^9) Log[2] Log[1-x1]-25084108800 (x1^10) Log[2] Log[1-x1]+24936422400 (x1^11) Log[2] Log[1-x1]-18532684800 (x1^12) Log[2] Log[1-x1]+10326873600 (x1^13) Log[2] Log[1-x1]-4282675200 (x1^14) Log[2] Log[1-x1]+1296806400 (x1^15) Log[2] Log[1-x1]-276710400 (x1^16) Log[2] Log[1-x1]+39052800 (x1^17) Log[2] Log[1-x1]-3225600 (x1^18) Log[2] Log[1-x1]+115200 (x1^19) Log[2] Log[1-x1]+38707200 (x1^5) ((Log[1-x1])^2)-387072000 (x1^6) ((Log[1-x1])^2)+1774080000 (x1^7) ((Log[1-x1])^2)-4935168000 (x1^8) ((Log[1-x1])^2)+9299520000 (x1^9) ((Log[1-x1])^2)-12542054400 (x1^10) ((Log[1-x1])^2)+12468211200 (x1^11) ((Log[1-x1])^2)-9266342400 (x1^12) ((Log[1-x1])^2)+5163436800 (x1^13) ((Log[1-x1])^2)-2141337600 (x1^14) ((Log[1-x1])^2)+648403200 (x1^15) ((Log[1-x1])^2)-138355200 (x1^16) ((Log[1-x1])^2)+19526400 (x1^17) ((Log[1-x1])^2)-1612800 (x1^18) ((Log[1-x1])^2)+57600 (x1^19) ((Log[1-x1])^2)-15482880 Log[2-x1]+135475200 x1 Log[2-x1]-541900800 (x1^2) Log[2-x1]+1306368000 (x1^3) Log[2-x1]-2125286400 (x1^4) Log[2-x1]+2657237760 (x1^5) Log[2-x1]-3490222080 (x1^6) Log[2-x1]+6027221760 (x1^7) Log[2-x1]-10120792320 (x1^8) Log[2-x1]+12545581440 (x1^9) Log[2-x1]-10099107840 (x1^10) Log[2-x1]+3950251200 (x1^11) Log[2-x1]+1417665600 (x1^12) Log[2-x1]-3241036200 (x1^13) Log[2-x1]+2419681200 (x1^14) Log[2-x1]-1095170400 (x1^15) Log[2-x1]+325363080 (x1^16) Log[2-x1]-62721960 (x1^17) Log[2-x1]+7297920 (x1^18) Log[2-x1]-429840 (x1^19) Log[2-x1]+7560 (x1^20) Log[2-x1]-77414400 (x1^5) Log[2] Log[2-x1]+774144000 (x1^6) Log[2] Log[2-x1]-3548160000 (x1^7) Log[2] Log[2-x1]+9870336000 (x1^8) Log[2] Log[2-x1]-18599040000 (x1^9) Log[2] Log[2-x1]+25084108800 (x1^10) Log[2] Log[2-x1]-24936422400 (x1^11) Log[2] Log[2-x1]+18532684800 (x1^12) Log[2] Log[2-x1]-10326873600 (x1^13) Log[2] Log[2-x1]+4282675200 (x1^14) Log[2] Log[2-x1]-1296806400 (x1^15) Log[2] Log[2-x1]+276710400 (x1^16) Log[2] Log[2-x1]-39052800 (x1^17) Log[2] Log[2-x1]+3225600 (x1^18) Log[2] Log[2-x1]-115200 (x1^19) Log[2] Log[2-x1]-77414400 (x1^5) Log[1-x1] Log[2-x1]+774144000 (x1^6) Log[1-x1] Log[2-x1]-3548160000 (x1^7) Log[1-x1] Log[2-x1]+9870336000 (x1^8) Log[1-x1] Log[2-x1]-18599040000 (x1^9) Log[1-x1] Log[2-x1]+25084108800 (x1^10) Log[1-x1] Log[2-x1]-24936422400 (x1^11) Log[1-x1] Log[2-x1]+18532684800 (x1^12) Log[1-x1] Log[2-x1]-10326873600 (x1^13) Log[1-x1] Log[2-x1]+4282675200 (x1^14) Log[1-x1] Log[2-x1]-1296806400 (x1^15) Log[1-x1] Log[2-x1]+276710400 (x1^16) Log[1-x1] Log[2-x1]-39052800 (x1^17) Log[1-x1] Log[2-x1]+3225600 (x1^18) Log[1-x1] Log[2-x1]-115200 (x1^19) Log[1-x1] Log[2-x1]+38707200 (x1^5) ((Log[2-x1])^2)-387072000 (x1^6) ((Log[2-x1])^2)+1774080000 (x1^7) ((Log[2-x1])^2)-4935168000 (x1^8) ((Log[2-x1])^2)+9299520000 (x1^9) ((Log[2-x1])^2)-12542054400 (x1^10) ((Log[2-x1])^2)+12468211200 (x1^11) ((Log[2-x1])^2)-9266342400 (x1^12) ((Log[2-x1])^2)+5163436800 (x1^13) ((Log[2-x1])^2)-2141337600 (x1^14) ((Log[2-x1])^2)+648403200 (x1^15) ((Log[2-x1])^2)-138355200 (x1^16) ((Log[2-x1])^2)+19526400 (x1^17) ((Log[2-x1])^2)-1612800 (x1^18) ((Log[2-x1])^2)+57600 (x1^19) ((Log[2-x1])^2)-77414400 (x1^5) Log[2] Log[x1]+774144000 (x1^6) Log[2] Log[x1]-3548160000 (x1^7) Log[2] Log[x1]+9870336000 (x1^8) Log[2] Log[x1]-18599040000 (x1^9) Log[2] Log[x1]+25084108800 (x1^10) Log[2] Log[x1]-24936422400 (x1^11) Log[2] Log[x1]+18532684800 (x1^12) Log[2] Log[x1]-10326873600 (x1^13) Log[2] Log[x1]+4282675200 (x1^14) Log[2] Log[x1]-1296806400 (x1^15) Log[2] Log[x1]+276710400 (x1^16) Log[2] Log[x1]-39052800 (x1^17) Log[2] Log[x1]+3225600 (x1^18) Log[2] Log[x1]-115200 (x1^19) Log[2] Log[x1]-77414400 (x1^5) Log[1-x1] Log[x1]+774144000 (x1^6) Log[1-x1] Log[x1]-3548160000 (x1^7) Log[1-x1] Log[x1]+9870336000 (x1^8) Log[1-x1] Log[x1]-18599040000 (x1^9) Log[1-x1] Log[x1]+25084108800 (x1^10) Log[1-x1] Log[x1]-24936422400 (x1^11) Log[1-x1] Log[x1]+18532684800 (x1^12) Log[1-x1] Log[x1]-10326873600 (x1^13) Log[1-x1] Log[x1]+4282675200 (x1^14) Log[1-x1] Log[x1]-1296806400 (x1^15) Log[1-x1] Log[x1]+276710400 (x1^16) Log[1-x1] Log[x1]-39052800 (x1^17) Log[1-x1] Log[x1]+3225600 (x1^18) Log[1-x1] Log[x1]-115200 (x1^19) Log[1-x1] Log[x1]+77414400 (x1^5) Log[2-x1] Log[x1]-774144000 (x1^6) Log[2-x1] Log[x1]+3548160000 (x1^7) Log[2-x1] Log[x1]-9870336000 (x1^8) Log[2-x1] Log[x1]+18599040000 (x1^9) Log[2-x1] Log[x1]-25084108800 (x1^10) Log[2-x1] Log[x1]+24936422400 (x1^11) Log[2-x1] Log[x1]-18532684800 (x1^12) Log[2-x1] Log[x1]+10326873600 (x1^13) Log[2-x1] Log[x1]-4282675200 (x1^14) Log[2-x1] Log[x1]+1296806400 (x1^15) Log[2-x1] Log[x1]-276710400 (x1^16) Log[2-x1] Log[x1]+39052800 (x1^17) Log[2-x1] Log[x1]-3225600 (x1^18) Log[2-x1] Log[x1]+115200 (x1^19) Log[2-x1] Log[x1]-115200 (((-2+x1))^4) (((-1+x1))^6) (x1^5) (42-84 x1+56 (x1^2)-14 (x1^3)+(x1^4)) PolyLog[2,1-x1]+115200 (((-2+x1))^4) (((-1+x1))^6) (x1^5) (42-84 x1+56 (x1^2)-14 (x1^3)+(x1^4)) PolyLog[2,1-(x1/2)]),{x1,0,1}]//Simplify

v5.1 IntegrateWithAssumptions (x1, x2, x3, y1, y2, y3)

Simplify`PiecewiseIntegrate;
PiecewiseDump`$ShowIntegrals=True;
Length[i=Integrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)]^3,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,x1},{y2,0,x2},{y3,0,x3}]]//Timing

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.HoldForm[Integrate[x__,opts___?OptionQ]]:>IntegrateWithAssumptions[x],3600])//Timing,
AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,Length[i]}
]

ReleaseHold[i[[11]]]//Timing

l[[11]]=((1364281-1968240 Log[2])/33264000);

ReleaseHold[i[[43]]]//Timing

l[[43]]=((1364281-1968240 Log[2])/33264000);

8FullSimplify[Plus@@l]

N[31/9000,10]

v6 IntegrateWithAssumptions (x1,x2,x3,y1,y2,y3)

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.HoldForm[Integrate[x__,opts___?OptionQ]]:>IntegrateWithAssumptions[x],600])//Timing,
AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,Length[i]}
]

v6 IntegrateWithAssumptions (x1, y1, x2, y2, x3, y3)

Simplify`PiecewiseIntegrate;
PiecewiseDump`$ShowIntegrals=True;
Length[i=Integrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)]^3,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}]]//Timing

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.HoldForm[Integrate[x__,opts___?OptionQ]]:>IntegrateWithAssumptions[x],600])//Timing,AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,Length[i]}
]

...crash...

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.HoldForm[Integrate[x__,opts___?OptionQ]]:>IntegrateWithAssumptions[x],600])//Timing,AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,18,18}
]

4

(2^4)NIntegrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)]^4,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3},Method->QuasiMonteCarlo,MaxPoints->10^6]//Timing

(2^4)Integrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^4,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}]//Timing

N[1/450]

5

v5.1 IntegrateWithAssumptions (x1, x2, x3, y1, y2, y3)

Simplify`PiecewiseIntegrate;
PiecewiseDump`$ShowIntegrals=True;
Length[i=Integrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)]^5,{x1,0,1},{x2,0,1},{x3,0,1},{y1,0,x1},{y2,0,x2},{y3,0,x3}]]//Timing

l={};
Do[
Print[{n,(v=TimeConstrained[i[[n]]/.HoldForm[Integrate[x__,opts___?OptionQ]]:>IntegrateWithAssumptions[x],3600])//Timing,
AppendTo[l,v];
If[FreeQ[v,(Integrate|$Aborted)],N[v],$Failed]}],
{n,Length[i]}
]

ReleaseHold[i[[11]]]//Timing

l[[11]]=((571602431-824647992 Log[2])/2542700160);

ReleaseHold[i[[43]]]//Timing

l[[43]]=((571602431-824647992 Log[2])/2542700160);

FullSimplify[(2^5)Plus@@l]

N[%]

(2^5)NIntegrate[Abs[(-x2 y1 + x3 y1 + x1 y2 - x3 y2 - x1 y3 + x2 y3)]^5,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3},Method->QuasiMonteCarlo,MaxPoints->10^6]//Timing

6

(2^6)Integrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^6,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}]//Timing

8

(2^8)Integrate[(-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3)^8,{x1,0,1},{y1,0,x1},{x2,0,1},{y2,0,x2},{x3,0,1},{y3,0,x3}]//Timing

Central Moments

(muc=Table[Integrate[(x-1/12)^n p[x],{x,0,1}],{n,0,25}]//FullSimplify)//Timing