Cube Line Picking—Face and Face

Author

Eric W. Weisstein
May 24, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Adjacent Faces

Monte Carlo

pts=Norm/@Table[{RandomReal[],RandomReal[],0}-{0,RandomReal[],RandomReal[]},{(10^6)}];//Timing

Mean[pts]

{Histogram[pts],Histogram[pts,{.02}],Histogram[pts,{.05}]}

Probability

(∫_((cos^(-1))((1/(Sqrt[u-1]))))^(π/4))(Sqrt[u-(sec^2)(θ)])\[DifferentialD]θ==2 (tan^(-1))((Sqrt[u-2]))+(Sqrt[u]) ((csc^(-1))((Sqrt[2-(2/u)]))-(tan^(-1))((Sqrt[(u-2) u])))-(π/2)

(∫_0^((cos^(-1))((1/(Sqrt[u])))))(Sqrt[u-(sec^2)(θ)])\[DifferentialD]θ==(1/2) π ((Sqrt[u])-1)

Integrate[(Sqrt[u-((Sec[θ])^2)]),{θ,0,ArcCos[(u^(-1/2))]},Assumptions->1<u<2]

j[u_?NumericQ]:=NIntegrate[(Sqrt[u-((Sec[θ])^2)]),{θ,0,ArcCos[(u^(-1/2))]}]

Plot[j[u],{u,1,2}]

Simplify[(1/2) π (1+(Sqrt[u]))-π]

Plot[(1/2) π (1+(Sqrt[u]))-j[u]-π,{u,1,2}]

Pl

i=Integrate[(Sqrt[u-((Sec[θ])^2)]),θ]

Limit[i,θ->0]

Limit[i,θ->ArcCos[(u^(-1/2))]]

FullSimplify[-((π (Sqrt[-(Sqrt[((-1+u)/u)]) (u^(3/2))]) (2 (Sqrt[-(((-1+u) Csc[2 ArcCos[(1/(Sqrt[u]))]])/(u^2))])+(Sqrt[-1+u]) (Sqrt[-((u Sin[2 ArcCos[(1/(Sqrt[u]))]])/(-1+u))])))/(2 (Sqrt[2]) (Sqrt[-1+u]))),1<u<2]

q[u_?NumericQ]:=NIntegrate[Sqrt[u - Sec[θ]^2], {θ, ArcCos[1/Sqrt[u - 1]], Pi/4}]

Plot[Subtract@@{q[u],2 ArcTan[(Sqrt[u-2])]+(Sqrt[u]) (ArcCsc[(Sqrt[2-(2/u)])]-ArcTan[(Sqrt[(u-2) u])])-(π/2)},{u,2,3}]

FullSimplify[p[l],1<l<(Sqrt[3])]

f[u_]:=Piecewise[{
{(π/2)(Sqrt[u])-(π/4)u,0<u<1},
{(π/4)+(π/4)u-(Sqrt[u-1])-(1/2)u(ArcSin[(u^(-1/2))]-ArcCos[(u^(-1/2))])-2(1/2) π ((Sqrt[u])-1),1<u<2},
{-1+(((u-2))^(1/2))+(1/2)(u-1)(ArcSin[(((u-1))^(-1/2))]-ArcCos[(((u-1))^(-1/2))])+(ArcSin[(((u-1))^(-1/2))]-ArcCos[(((u-1))^(-1/2))])-2(2ArcTan[(Sqrt[u-2])]+(Sqrt[u]) (ArcCsc[(Sqrt[2-(2/u)])]-ArcTan[(Sqrt[(u-2) u])])-(π/2)),2<u<3}
}]
p[u_]:=2u f[u^2]

FullSimplify[p[l],0<l<(Sqrt[3])]

Plot[Subtract@@{q[u],2 ArcTan[(Sqrt[u-2])]+(Sqrt[u]) (ArcCsc[(Sqrt[2-(2/u)])]-ArcTan[(Sqrt[(u-2) u])])-(π/2)},{u,2,3}]

Plot[p[u],{u,0,(Sqrt[3])},Exclusions->None,PlotRange->All]

Plot[p[u],{u,1.3,1.5},Exclusions->None,PlotRange->All]

Plot

FullSimplify[Integrate[p[v],{v,0,u},Assumptions->0<u<(Sqrt[3])],0<u<(Sqrt[3])]

d[u_]:=Piecewise[{{(1/24)*Pi*(8 - 3*u)*u^3, u <= 1}, 
   {(1/24)*(4*(2 - 5*u^2)*Sqrt[-1 + u^2] + Pi*(-9 + 2*(15 - 8*u)*u^2) + 12*u^4*ArcSec[u]), 
    Inequality[1, Less, u, LessEqual, Sqrt[2]]}}, 
  (1/24)*(16 - 51*Pi + 20*Sqrt[-2 + u^2] + 20*u^2*Sqrt[-2 + u^2] - 
    32*u^3*ArcCsc[Sqrt[2 - 2/u^2]] + 12*(-1 + 2*u^2)*ArcCsc[Sqrt[-1 + u^2]] + 
    3*u^4*(Pi - 4*ArcSec[Sqrt[-1 + u^2]]) - 48*ArcTan[(-2 + u)/Sqrt[-2 + u^2]] + 
    48*ArcTan[(2 + u)/Sqrt[-2 + u^2]] + 6*u^2*(-4 + 3*Pi - 16*ArcTan[Sqrt[-2 + u^2]]) + 
    120*ArcTan[Sqrt[-2 + u^2]] + 32*u^3*ArcTan[u*Sqrt[-2 + u^2]])]

d[(Sqrt[3])]//Simplify

Integrate[p[v],{v,0,(Sqrt[3])}]

MakeBoxes[cdf[x_],TraditionalForm]:=RowBox[{"D","(",MakeBoxes[x,TraditionalForm],")"}]

GraphicsGrid[{{Plot[p[u],{u,1,(Sqrt[3])},PlotStyle->{Red},Exclusions->None,AxesLabel->{l,P[l]}],
Plot[d[u],{u,1,(Sqrt[3])},PlotStyle->{Red},Exclusions->None,AxesLabel->{l,cdf[l]//TraditionalForm}]}},ImageSize->400,Spacings->-30]

Moments

NIntegrate[p[u],{u,0,(Sqrt[3])}]

Mean

Integrate[u p[u],{u,0,(Sqrt[3])}]//FullSimplify

(1/180) (-11 π+6 (7 (Sqrt[2])-(Sqrt[3])+3 ArcSinh[1]+84 Log[2]-168 Log[1+(Sqrt[3])]+50 Log[7+4 (Sqrt[3])]))//N

Expand[(((2+(Sqrt[3])))^2)]

N[(1/180) (-11 π+6 (7 (Sqrt[2])-(Sqrt[3])+3 ArcSinh[1]+84 Log[2]-168 Log[1+(Sqrt[3])]+100 Log[2+(Sqrt[3])])),10]

FullSimplify[(1/180) (-11 π+6 (7 (Sqrt[2])-(Sqrt[3])+3 ArcSinh[1]+84 Log[2]-168 Log[1+(Sqrt[3])]+100 Log[2+(Sqrt[3])]))]//Together

Any

Integrate[(u^n)p[u],{u,0,(Sqrt[3])},Assumptions->n>0]

Odd

Table[Integrate[(u^(2n-1)) p[u],{u,0,(Sqrt[3])}],{n,5}]//FullSimplify//Together//Column//TraditionalForm

Even

Table[Integrate[(u^(2n)) p[u],{u,0,(Sqrt[3])}],{n,0,5}]//Timing

Opposite Faces

Monte Carlo

pts=Norm/@Table[{RandomReal[],RandomReal[],0}-{RandomReal[],RandomReal[],1},{(10^6)}];//Timing

Mean[pts]

{Histogram[pts],Histogram[pts,{.02}],Histogram[pts,{.05}]}

Probability

f[u_]:=Piecewise[{
{π-4(Sqrt[u-1])+u-1,1<u<2},
{-2-(u-1)+4(Sqrt[u-2])+2(ArcSin[(((u-1))^(-1/2))]-ArcCos[(((u-1))^(-1/2))]),2<u<3}
}]
p[u_]:=2u f[u^2]

FullSimplify[p[l],1<l<(Sqrt[3])]

p[u_]:=2 u ((Piecewise[{{-1+π+(u^2)-4 (Sqrt[-1+(u^2)]), (u^2)<2}, {-1+π-(u^2)+4 (Sqrt[-2+(u^2)])-4 ArcSec[(Sqrt[-1+(u^2)])], (u^2)>2}, {0, True}}]))

Plot[p[u],{u,1,(Sqrt[3])}]

Plot

FullSimplify[Integrate[p[v],{v,1,u},Assumptions->1<u<(Sqrt[3])],1<u<(Sqrt[3])]

d[u_]:=Piecewise[{{(-(1/6))*(-1 + u^2)*(3 - 6*Pi - 3*u^2 + 
      16*Sqrt[-1 + u^2]), u <= Sqrt[2]}}, 
  (1/6)*(11 - 6*Pi - 3*u^4 - 8*Sqrt[-2 + u^2] + 
    16*u^2*Sqrt[-2 + u^2] + 
    6*u^2*(-1 + Pi - 4*ArcSec[Sqrt[-1 + u^2]]) + 
    24*ArcTan[Sqrt[-2 + u^2]])]

MakeBoxes[cdf[x_],TraditionalForm]:=RowBox[{"D","(",MakeBoxes[x,TraditionalForm],")"}]

GraphicsGrid[{{Plot[p[u],{u,1,(Sqrt[3])},PlotStyle->{Red},Exclusions->None,AxesLabel->{l,P[l]}],
Plot[d[u],{u,1,(Sqrt[3])},PlotStyle->{Red},Exclusions->None,AxesLabel->{l,cdf[l]//TraditionalForm}]}},ImageSize->400,Spacings->-30]

Moments

Integrate[p[u],{u,1,(Sqrt[3])}]

Mean

Integrate[u p[u],{u,1,(Sqrt[3])}]//FullSimplify

N[%]

(1/4)(((1+(Sqrt[3])))^4)//Expand

-((2 π)/9)+ArcSinh[1]+Log[4]+(1/15) (4+3 (Sqrt[2])-4 (Sqrt[3])-60 Log[1+(Sqrt[3])]+20 Log[(1/4)(((1+(Sqrt[3])))^4)])//N

N[-((2 π)/9)+ArcSinh[1]+Log[4]+(1/15) (4+3 (Sqrt[2])-4 (Sqrt[3])-60 Log[1+(Sqrt[3])]+20Log[(1/4)]+80 Log[(1+(Sqrt[3]))]),10]

FullSimplify[-((2 π)/9)+ArcSinh[1]+Log[4]+(1/15) (4+3 (Sqrt[2])-4 (Sqrt[3])-60 Log[1+(Sqrt[3])]+20Log[(1/4)]+80 Log[(1+(Sqrt[3]))])]

MAppellF1[x_,y_,z_,a_,b_,c_]:=AppellF1[x,y,z,a,b,c](b^(z+x-1))

-(2/3)π+((Sqrt[2])/5)-(8/5)(Sqrt[3])+(4/15)-4Log[2]-Log[(Sqrt[2])-1]+4Log[(Sqrt[3])-1]+(16/3)AppellF1[(1/2),-(3/2),2,(3/2),(1/4),(1/2)]//N

-(2/3)π+((Sqrt[2])/5)-(8/5)(Sqrt[3])+(4/15)-4Log[2]-Log[(Sqrt[2])-1]+4Log[(Sqrt[3])-1]+(16/3)MAppellF1[(1/2),-(3/2),2,(3/2),(1/4),(1/2)]//N

Any

Integrate[(u^n)p[u],{u,1,(Sqrt[3])},Assumptions->n>0]

Integrate[2(u^(n+1))(-1+π+(u^2)-4 (Sqrt[-1+(u^2)])),{u,1,(Sqrt[2])}]

Integrate[2(u^(n+1))(-1+π-(u^2)+4 (Sqrt[-2+(u^2)])-4 ArcSec[(Sqrt[-1+(u^2)])]),{u,(Sqrt[2]),(Sqrt[3])}]

Integrate[2(u^(n+1))(-1+π-(u^2)+4 (Sqrt[-2+(u^2)])-4 ArcSec[(Sqrt[-1+(u^2)])]),u]

Odd

Table[Integrate[(u^(2n-1)) p[u],{u,1,(Sqrt[3])}],{n,5}]//FullSimplify//Together//Column//TraditionalForm

N[Last[%]]

Even

Table[Integrate[(u^(2n)) p[u],{u,1,(Sqrt[3])}],{n,0,5}]//Timing

Different Faces

Monte Carlo

FacePoint[1]:={0,Random[Real],Random[Real]}
FacePoint[2]:={1,Random[Real],Random[Real]}
FacePoint[3]:={Random[Real],0,Random[Real]}
FacePoint[4]:={Random[Real],1,Random[Real]}
FacePoint[5]:={Random[Real],Random[Real],0}
FacePoint[6]:={Random[Real],Random[Real],1}

<<DiscreteMath`Combinatorica`

f=KSubsets[Range[6],2]

Mean[Norm/@Subtract@@@Map[FacePoint,f[[#]]&/@Table[Random[Integer,{1,15}],{10^6}],{2}]]//Timing

First Integral: (∫_0^1)(∫_0^1)(∫_0^1)(∫_0^1)(Sqrt[(((z-w))^2)+(x^2)+(y^2)])dzdwdydx

Numerical

4/5NIntegrate[Sqrt[x^2+y^2+(z-w)^2],{x,0,1},{y,0,1},{w,0,1},{z,0,1}]//Timing

4/5NIntegrate[Sqrt[x^2+y^2+(z-w)^2],{x,0,1},{y,0,1},{w,0,1},{z,0,1},Method->GaussKronrod]//Timing

4/5NIntegrate[Sqrt[x^2+y^2+(z-w)^2],{x,0,1},{y,0,1},{w,0,1},{z,0,1},Method->MultiDimensional]//Timing

v4.2.1

4/5NIntegrate[Sqrt[x^2+y^2+(z-w)^2],{x,0,1},{y,0,1},{w,0,1},{z,0,1},WorkingPrecision->20]//Timing

v5

4/5NIntegrate[Sqrt[x^2+y^2+(z-w)^2],{x,0,1},{y,0,1},{w,0,1},{z,0,1},WorkingPrecision->20]//Timing

Default Options

Timing[i=(4/5)Integrate[Sqrt[x^2+y^2+(z-w)^2],{x,0,1},{y,0,1},{w,0,1},{z,0,1}]]

N[i,20]

i1=FullSimplify[i]

N[(1/1800)(-88 π-99 Log[2]+6 (56 (Sqrt[2])-8 (Sqrt[3])+63 ArcCsch[(Sqrt[2])]+24 ArcSinh[1]-80 Log[-1+(Sqrt[3])]+113 Log[1+(Sqrt[3])]))]

v6

(i1=4/5Integrate[Sqrt[x^2+y^2+(z-w)^2],{x,0,1},{y,0,1},{w,0,1},{z,0,1}])//Timing

N[(1/900) (168 (Sqrt[2])-24 (Sqrt[3])-44 π+72 ArcSinh[1]+162 ArcSinh[(1/(Sqrt[2]))]+24 Log[2]-240 Log[-1+(Sqrt[3])]+192 Log[1+(Sqrt[3])]+20 Log[26+15 (Sqrt[3])]+3 Log[70226+40545 (Sqrt[3])])]

FullSimplify[(1/900) (168 (Sqrt[2])-24 (Sqrt[3])-44 π+72 ArcSinh[1]+162 ArcSinh[(1/(Sqrt[2]))]+24 Log[2]-240 Log[-1+(Sqrt[3])]+192 Log[1+(Sqrt[3])]+20 Log[26+15 (Sqrt[3])]+3 Log[70226+40545 (Sqrt[3])])]

N[%]

Second Integral: (∫_0^1)(∫_0^1)(∫_0^1)(∫_0^1)(Sqrt[(((y-u))^2)+(((z-w))^2)+1])dudwdydz

Numerical

(1/5)NIntegrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1}]//Timing

1/5NIntegrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1},Method->GaussKronrod]//Timing

1/5NIntegrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1},Method->MultiDimensional]//Timing

(1/5)NIntegrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1},WorkingPrecision->15]//Timing

(1/5)NIntegrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1},WorkingPrecision->20]//Timing

...hang...

GenerateConditions->Automatic

v5.0.1

Timing[i2=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1}]]

N[%]

...wrong...

v5.1

Timing[i2=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1}]]

N[%]

v5.2

Timing[i2=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1}]]

...crash...

v6

v6 old

$Version

Timing[i=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},Assumptions->0<u<1&&0<w<1];]

i

Timing[i2=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{u,0,1},{w,0,1},{y,0,1},{z,0,1}]]

3861.2 Second

i2=(-5*u*Sqrt[1+u^2]-2*u^3*Sqrt[1+u^2]+10*u*Sqrt[2+u^2]+2*u^3*Sqrt[2+u^2]-7*Sqrt[2-2*u+u^2]+11*u*Sqrt[2-2*u+u^2]-6*u^2*Sqrt[2-2*u+u^2]+2*u^3*Sqrt[2-2*u+u^2]+12*Sqrt[3-2*u+u^2]-16*u*Sqrt[3-2*u+u^2]+6*u^2*Sqrt[3-2*u+u^2]-2*u^3*Sqrt[3-2*u+u^2]-3*ArcSinh[1-u]+12*ArcSinh[(1-u)/Sqrt[2]]-3*ArcSinh[u]+12*ArcSinh[u/Sqrt[2]])/48+(-5*u*Sqrt[1+u^2]-2*u^3*Sqrt[1+u^2]+2*u*Sqrt[2+u^2]+2*u^3*Sqrt[2+u^2]-7*Sqrt[2-2*u+u^2]+11*u*Sqrt[2-2*u+u^2]-6*u^2*Sqrt[2-2*u+u^2]+2*u^3*Sqrt[2-2*u+u^2]+4*Sqrt[3-2*u+u^2]-8*u*Sqrt[3-2*u+u^2]+6*u^2*Sqrt[3-2*u+u^2]-2*u^3*Sqrt[3-2*u+u^2]-7*ArcSinh[1-u]-7*ArcSinh[u]-(2+4*I)*Log[-24/5-(12*I)/5]-(2-4*I)*Log[-24/5+(12*I)/5]+(2-4*I)*Log[24/5-(12*I)/5]+(2+4*I)*Log[24/5+(12*I)/5]-6*u*Log[1+u^2]-2*u^3*Log[1+u^2]-8*Log[2-2*u+u^2]+12*u*Log[2-2*u+u^2]-6*u^2*Log[2-2*u+u^2]+2*u^3*Log[2-2*u+u^2]+4*Log[1+u^2+u*Sqrt[1+u^2]]+12*u*Log[-1+Sqrt[2+u^2]]+4*u^3*Log[-1+Sqrt[2+u^2]]-(2+4*I)*Log[(6/5+(18*I)/5)*((1-I)+u^2+u*Sqrt[2+u^2])]-(2-4*I)*Log[(6/5-(18*I)/5)*((1+I)+u^2+u*Sqrt[2+u^2])]+16*Log[-1+Sqrt[3-2*u+u^2]]-24*u*Log[-1+Sqrt[3-2*u+u^2]]+12*u^2*Log[-1+Sqrt[3-2*u+u^2]]-4*u^3*Log[-1+Sqrt[3-2*u+u^2]]-4*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(2+4*I)*Log[(-6/5-(18*I)/5)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(2-4*I)*Log[(-6/5+(18*I)/5)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))])/48+(4+2*u*Sqrt[1+u^2]-2*u*Sqrt[2+u^2]+2*Sqrt[2-2*u+u^2]-2*u*Sqrt[2-2*u+u^2]-2*Sqrt[3-2*u+u^2]+2*u*Sqrt[3-2*u+u^2]-2*ArcSinh[u]-4*ArcTan[1-u]-4*ArcTan[u]-2*Log[1+u^2]+4*Log[2-2*u+u^2]+4*Log[1+u^2+u*Sqrt[1+u^2]]-4*u*Log[-1+Sqrt[2+u^2]]-(1-I)*Log[(1+I)+I*u^2+I*u*Sqrt[2+u^2]]-(1+I)*u*Log[(1+I)+I*u^2+I*u*Sqrt[2+u^2]]-(2*I)*Log[(-1/2-I/2)*((1-I)+u^2+u*Sqrt[2+u^2])]+(1+3*I)*Log[(-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+(1+I)*u*Log[(-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+(2*I)*Log[(-1/2+I/2)*((1+I)+u^2+u*Sqrt[2+u^2])]-(1+I)*Log[(-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-(1-I)*u*Log[(-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]+(1-3*I)*Log[(2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]+(1-I)*u*Log[(2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-2*Log[(2-2*u+u^2)*(1-u+Sqrt[2-2*u+u^2])]-4*Log[-1+Sqrt[3-2*u+u^2]]+4*u*Log[-1+Sqrt[3-2*u+u^2]]-4*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(1-I)*Log[(1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2])]+(1+I)*u*Log[(1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2])]+(2*I)*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(1+3*I)*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(1+I)*u*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(2*I)*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(1+I)*Log[(-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(1-I)*u*Log[(-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(1-3*I)*Log[(2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(1-I)*u*Log[(2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))])/8+(1/8+I/8)*u^2*((2-2*I)+(1-I)*u*Sqrt[1+u^2]-(1-I)*u*Sqrt[2+u^2]+(1-I)*Sqrt[2-2*u+u^2]-(1-I)*u*Sqrt[2-2*u+u^2]-(1-I)*Sqrt[3-2*u+u^2]+(1-I)*u*Sqrt[3-2*u+u^2]-(1-I)*ArcSinh[u]-(2-2*I)*ArcTan[1-u]-(2-2*I)*ArcTan[u]-(1-I)*Log[1+u^2]+(1-I)*Log[2-2*u+u^2]+(2-2*I)*Log[1+u^2+u*Sqrt[1+u^2]]-(2-2*I)*u*Log[-1+Sqrt[2+u^2]]+I*Log[(1+I)+I*u^2+I*u*Sqrt[2+u^2]]-u*Log[(1+I)+I*u^2+I*u*Sqrt[2+u^2]]-(1+I)*Log[(-1/2-I/2)*((1-I)+u^2+u*Sqrt[2+u^2])]+(2+I)*Log[(-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+u*Log[(-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+(1+I)*Log[(-1/2+I/2)*((1+I)+u^2+u*Sqrt[2+u^2])]-Log[(-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]+I*u*Log[(-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-(1+2*I)*Log[(2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-I*u*Log[(2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-(1-I)*Log[1-u+Sqrt[2-2*u+u^2]]-(2-2*I)*Log[-1+Sqrt[3-2*u+u^2]]+(2-2*I)*u*Log[-1+Sqrt[3-2*u+u^2]]-(2-2*I)*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]-I*Log[(1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2])]+u*Log[(1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2])]+(1+I)*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(2+I)*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-u*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(1+I)*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+Log[(-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-I*u*Log[(-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(1+2*I)*Log[(2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+I*u*Log[(2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))])+(4*u*Sqrt[2+u^2]+4*Sqrt[3-2*u+u^2]-4*u*Sqrt[3-2*u+u^2]+8*ArcSinh[1-u]+8*ArcSinh[u]+(4+2*I)*Log[-6/5-(12*I)/5]+(4-2*I)*Log[-6/5+(12*I)/5]-(4-2*I)*Log[6/5-(12*I)/5]-(4+2*I)*Log[6/5+(12*I)/5]+3*u*Log[1+u^2]+u^3*Log[1+u^2]+4*Log[2-2*u+u^2]-6*u*Log[2-2*u+u^2]+3*u^2*Log[2-2*u+u^2]-u^3*Log[2-2*u+u^2]-8*Log[1+u^2+u*Sqrt[1+u^2]]-6*u*Log[-1+Sqrt[2+u^2]]-2*u^3*Log[-1+Sqrt[2+u^2]]+(4+2*I)*Log[(9/5+(3*I)/5)*((1+I)+I*u^2+I*u*Sqrt[2+u^2])]+(4-2*I)*Log[(-3/5-(9*I)/5)*((1+I)+u^2+u*Sqrt[2+u^2])]-8*Log[-1+Sqrt[3-2*u+u^2]]+12*u*Log[-1+Sqrt[3-2*u+u^2]]-6*u^2*Log[-1+Sqrt[3-2*u+u^2]]+2*u^3*Log[-1+Sqrt[3-2*u+u^2]]+8*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]-(4+2*I)*Log[(3/5-(9*I)/5)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(4-2*I)*Log[(3/5+(9*I)/5)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))])/24-(u^2*(4-2*u*Sqrt[1+u^2]+2*u*Sqrt[2+u^2]-2*Sqrt[2-2*u+u^2]+2*u*Sqrt[2-2*u+u^2]+2*Sqrt[3-2*u+u^2]-2*u*Sqrt[3-2*u+u^2]+2*ArcSinh[u]-4*ArcTan[1-u]-4*ArcTan[u]+Log[256]-4*Log[(-1+u)^(-2)]-2*u*Log[1+u^2]+2*u*Log[2-2*u+u^2]-4*Log[1+Sqrt[3+(-2+u)*u]]-4*Log[4*(1+u^2+u*Sqrt[1+u^2])]+4*Log[-u+Sqrt[2+u^2]]-(2*I)*Log[-((2-I*u+Sqrt[2+u^2])/(I+u)^2)]-2*u*Log[-((2-I*u+Sqrt[2+u^2])/(I+u)^2)]+(2*I)*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^2)]-2*u*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^2)]+Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+I*u*Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-I*u*Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-I*u*Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]+Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]+I*u*Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]+2*Log[1-u+Sqrt[2-2*u+u^2]]-4*Log[1-u+Sqrt[3-2*u+u^2]]+(2*I)*Log[((2+I)-I*u+Sqrt[3-2*u+u^2])/((1+I)+(1-2*I)*u-u^2)]+2*u*Log[((2+I)-I*u+Sqrt[3-2*u+u^2])/((1+I)+(1-2*I)*u-u^2)]-(2*I)*Log[((2-I)+I*u+Sqrt[3-2*u+u^2])/((1-I)+(1+2*I)*u-u^2)]+2*u*Log[((2-I)+I*u+Sqrt[3-2*u+u^2])/((1-I)+(1+2*I)*u-u^2)]+4*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]-Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-I*u*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+I*u*Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+I*u*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-I*u*Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/8+(-4+2*u*Sqrt[1+u^2]-2*u*Sqrt[2+u^2]+2*Sqrt[2-2*u+u^2]-2*u*Sqrt[2-2*u+u^2]-2*Sqrt[3-2*u+u^2]+2*u*Sqrt[3-2*u+u^2]-2*ArcSinh[1-u]-2*ArcSinh[u]+4*ArcTan[1-u]+4*ArcTan[u]-4*Log[4]-4*u*Log[4]+u*Log[256]+4*Log[(-1+u)^(-2)]+2*u*Log[1+u^2]-2*u*Log[2-2*u+u^2]+4*Log[1+Sqrt[3+(-2+u)*u]]+4*Log[1-u+Sqrt[3+(-2+u)*u]]+4*Log[4*(1+u^2+u*Sqrt[1+u^2])]-4*Log[-u+Sqrt[2+u^2]]+(2*I)*Log[-((2-I*u+Sqrt[2+u^2])/(I+u)^2)]+2*u*Log[-((2-I*u+Sqrt[2+u^2])/(I+u)^2)]-(2*I)*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^2)]+2*u*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^2)]-Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-I*u*Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+I*u*Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]+I*u*Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-I*u*Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-(2*I)*Log[((2+I)-I*u+Sqrt[3-2*u+u^2])/((1+I)+(1-2*I)*u-u^2)]-2*u*Log[((2+I)-I*u+Sqrt[3-2*u+u^2])/((1+I)+(1-2*I)*u-u^2)]+(2*I)*Log[((2-I)+I*u+Sqrt[3-2*u+u^2])/((1-I)+(1+2*I)*u-u^2)]-2*u*Log[((2-I)+I*u+Sqrt[3-2*u+u^2])/((1-I)+(1+2*I)*u-u^2)]-4*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]+Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+I*u*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-I*u*Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-I*u*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+I*u*Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))])/8-(u*(-6-12*u-8*Sqrt[1+u^2]+4*u^2*Sqrt[1+u^2]+4*Sqrt[2+u^2]-4*u^2*Sqrt[2+u^2]+16*Sqrt[2-2*u+u^2]-4*u*Sqrt[2-2*u+u^2]-4*u^2*Sqrt[2-2*u+u^2]-12*Sqrt[3-2*u+u^2]+4*u*Sqrt[3-2*u+u^2]+4*u^2*Sqrt[3-2*u+u^2]-12*u*ArcSinh[u]+24*u*ArcTan[1-u]+24*u*ArcTan[u]-24*u*Log[4]-12*Log[8]+6*Log[64]+6*u*Log[256]+24*u*Log[(-1+u)^(-2)]+48*u*Log[u]-6*Log[1+u^2]+6*u^2*Log[1+u^2]+6*Log[2-2*u+u^2]-6*u^2*Log[2-2*u+u^2]+12*Log[1+Sqrt[3+(-2+u)*u]]+24*u*Log[1+u^(-2)+Sqrt[1+u^2]/u]-24*u*Log[-u+Sqrt[2+u^2]]-6*Log[-((2-I*u+Sqrt[2+u^2])/(I+u)^3)]+(12*I)*u*Log[-((2-I*u+Sqrt[2+u^2])/(I+u)^3)]+6*u^2*Log[-((2-I*u+Sqrt[2+u^2])/(I+u)^3)]-6*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^3)]-(12*I)*u*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^3)]+6*u^2*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^3)]+(3*I)*Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-6*u*Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-(3*I)*u^2*Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-(3*I)*Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-6*u*Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+(3*I)*u^2*Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-(3*I)*Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-6*u*Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]+(3*I)*u^2*Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]+(3*I)*Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-6*u*Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-(3*I)*u^2*Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]-12*u*Log[1-u+Sqrt[2-2*u+u^2]]+24*u*Log[1-u+Sqrt[3-2*u+u^2]]+6*Log[-(((2+I)-I*u+Sqrt[3-2*u+u^2])/(((-1+I)+u)*(I+u)^2))]-(12*I)*u*Log[-(((2+I)-I*u+Sqrt[3-2*u+u^2])/(((-1+I)+u)*(I+u)^2))]-6*u^2*Log[-(((2+I)-I*u+Sqrt[3-2*u+u^2])/(((-1+I)+u)*(I+u)^2))]+6*Log[-(((2-I)+I*u+Sqrt[3-2*u+u^2])/(((-1-I)+u)*(-I+u)^2))]+(12*I)*u*Log[-(((2-I)+I*u+Sqrt[3-2*u+u^2])/(((-1-I)+u)*(-I+u)^2))]-6*u^2*Log[-(((2-I)+I*u+Sqrt[3-2*u+u^2])/(((-1-I)+u)*(-I+u)^2))]-24*u*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]-(3*I)*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+6*u*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(3*I)*u^2*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(3*I)*Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+6*u*Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(3*I)*u^2*Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(3*I)*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+6*u*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(3*I)*u^2*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(3*I)*Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+6*u*Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(3*I)*u^2*Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/24+(-2-3*u-6*u^2+Sqrt[2+(-2+u)*u]*(-2-(-2+u)*u+Sqrt[(2+(-2+u)*u)*(3+(-2+u)*u)]+Sqrt[1+(2+(-2+u)*u)^(-1)])+5*u*Sqrt[2+(-2+u)*u]*(-2-(-2+u)*u+Sqrt[(2+(-2+u)*u)*(3+(-2+u)*u)]+Sqrt[1+(2+(-2+u)*u)^(-1)])+5*u*(Sqrt[1+u^2]-2*Sqrt[2+u^2]+u^2*(Sqrt[1+u^2]-Sqrt[2+u^2]))+(3+8*u^3*(-Sqrt[1+u^2]+Sqrt[2+u^2])+4*u*(-5*Sqrt[1+u^2]+4*Sqrt[2+u^2])-12*ArcSinh[u])/16-6*ArcTan[1-u]+18*u^2*ArcTan[1-u]-6*ArcTan[u]+18*u^2*ArcTan[u]-9*u*Log[1+u^2]+3*u^3*Log[1+u^2]+9*u*Log[2-2*u+u^2]-3*u^3*Log[2-2*u+u^2]+6*(1+Sqrt[2+(-2+u)*u]-Sqrt[3+(-2+u)*u]+Log[1+Sqrt[3+(-2+u)*u]])-(9*(-1-2*Sqrt[2+(-2+u)*u]+2*u*Sqrt[2+(-2+u)*u]+2*Sqrt[3+(-2+u)*u]-2*u*Sqrt[3+(-2+u)*u]-2*ArcSinh[1-u]+4*Log[1-u+Sqrt[3+(-2+u)*u]]))/4+(9*u^2*(-1-2*Sqrt[2+(-2+u)*u]+2*u*Sqrt[2+(-2+u)*u]+2*Sqrt[3+(-2+u)*u]-2*u*Sqrt[3+(-2+u)*u]-2*ArcSinh[1-u]+4*Log[1-u+Sqrt[3+(-2+u)*u]]))/2+(9*(-1+2*u*Sqrt[1+u^2]-2*u*Sqrt[2+u^2]+2*ArcSinh[u]+4*Log[-u+Sqrt[2+u^2]]))/4-(9*u^2*(-1+2*u*Sqrt[1+u^2]-2*u*Sqrt[2+u^2]+2*ArcSinh[u]+4*Log[-u+Sqrt[2+u^2]]))/2+(9*u^2*(4*I+(3*I)*Pi+(4*I)*Sqrt[1+u^2]-(4*I)*Sqrt[2+u^2]+Log[256]+8*Log[u]+4*Log[1+u^(-2)+Sqrt[1+u^2]/u]+(4*I)*Log[(-12*(2-I*u+Sqrt[2+u^2]))/(I+u)^4]-2*Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-2*Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4-(9*u*(4+3*Pi+4*Sqrt[1+u^2]-4*Sqrt[2+u^2]-(4*I)*Log[4]-(8*I)*Log[u]-(4*I)*Log[1+u^(-2)+Sqrt[1+u^2]/u]+4*Log[(-12*(2-I*u+Sqrt[2+u^2]))/(I+u)^4]+(2*I)*Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+(2*I)*Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4+(3*u^3*(4+3*Pi+4*Sqrt[1+u^2]-4*Sqrt[2+u^2]-(4*I)*Log[4]-(8*I)*Log[u]-(4*I)*Log[1+u^(-2)+Sqrt[1+u^2]/u]+4*Log[(-12*(2-I*u+Sqrt[2+u^2]))/(I+u)^4]+(2*I)*Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+(2*I)*Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4+(3*(-4*I-(3*I)*Pi-(4*I)*Sqrt[1+u^2]+(4*I)*Sqrt[2+u^2]-4*Log[4]-8*Log[u]-4*Log[1+u^(-2)+Sqrt[1+u^2]/u]-(4*I)*Log[(-12*(2-I*u+Sqrt[2+u^2]))/(I+u)^4]+2*Log[(2+2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+2*Log[(-2+2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4+(9*u^2*(4*I+I*Pi-(4*I)*Sqrt[1+u^2]+(4*I)*Sqrt[2+u^2]+Log[256]+8*Log[u]+4*Log[1+u^(-2)+Sqrt[1+u^2]/u]-(4*I)*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^4)]-2*Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-2*Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4-(9*u*(-4-Pi+4*Sqrt[1+u^2]-4*Sqrt[2+u^2]+(4*I)*Log[4]+(8*I)*Log[u]+(4*I)*Log[1+u^(-2)+Sqrt[1+u^2]/u]+4*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^4)]-(2*I)*Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-(2*I)*Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4+(3*u^3*(-4-Pi+4*Sqrt[1+u^2]-4*Sqrt[2+u^2]+(4*I)*Log[4]+(8*I)*Log[u]+(4*I)*Log[1+u^(-2)+Sqrt[1+u^2]/u]+4*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^4)]-(2*I)*Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-(2*I)*Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4+(3*(-4*I-I*Pi+(4*I)*Sqrt[1+u^2]-(4*I)*Sqrt[2+u^2]-4*Log[4]-8*Log[u]-4*Log[1+u^(-2)+Sqrt[1+u^2]/u]+(4*I)*Log[-((2+I*u+Sqrt[2+u^2])/(-I+u)^4)]+2*Log[(-2-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+2*Log[(2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4+(-3-28*Sqrt[2-2*u+u^2]+24*Sqrt[3-2*u+u^2]+u*(44*Sqrt[2-2*u+u^2]-40*Sqrt[3-2*u+u^2])-24*u^2*(Sqrt[2-2*u+u^2]-Sqrt[3-2*u+u^2])+8*u^3*(Sqrt[2-2*u+u^2]-Sqrt[3-2*u+u^2])-12*Log[1-u+Sqrt[2-2*u+u^2]])/16+(3*(4*I+(3*I)*Pi+(4*I)*Sqrt[2-2*u+u^2]-(4*I)*Sqrt[3-2*u+u^2]+Log[256]-4*Log[(-1+u)^(-2)]+(4*I)*Log[(-12*((2+I)-I*u+Sqrt[3-2*u+u^2]))/(((-1+I)+u)*(I+u)^3)]+4*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]-2*Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-2*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4+(9*u*(4+3*Pi+4*Sqrt[2-2*u+u^2]-4*Sqrt[3-2*u+u^2]-(4*I)*Log[4]+(4*I)*Log[(-1+u)^(-2)]+4*Log[(-12*((2+I)-I*u+Sqrt[3-2*u+u^2]))/(((-1+I)+u)*(I+u)^3)]-(4*I)*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]+(2*I)*Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(2*I)*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4-(3*u^3*(4+3*Pi+4*Sqrt[2-2*u+u^2]-4*Sqrt[3-2*u+u^2]-(4*I)*Log[4]+(4*I)*Log[(-1+u)^(-2)]+4*Log[(-12*((2+I)-I*u+Sqrt[3-2*u+u^2]))/(((-1+I)+u)*(I+u)^3)]-(4*I)*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]+(2*I)*Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(2*I)*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4+(9*u^2*(-4*I-(3*I)*Pi-(4*I)*Sqrt[2-2*u+u^2]+(4*I)*Sqrt[3-2*u+u^2]-4*Log[4]+4*Log[(-1+u)^(-2)]-(4*I)*Log[(-12*((2+I)-I*u+Sqrt[3-2*u+u^2]))/(((-1+I)+u)*(I+u)^3)]-4*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]+2*Log[(2+2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+2*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4+(3*(4*I+I*Pi-(4*I)*Sqrt[2-2*u+u^2]+(4*I)*Sqrt[3-2*u+u^2]+Log[256]-4*Log[(-1+u)^(-2)]-(4*I)*Log[-(((2-I)+I*u+Sqrt[3-2*u+u^2])/(((-1-I)+u)*(-I+u)^3))]+4*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]-2*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-2*Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4+(9*u*(-4-Pi+4*Sqrt[2-2*u+u^2]-4*Sqrt[3-2*u+u^2]+(4*I)*Log[4]-(4*I)*Log[(-1+u)^(-2)]+4*Log[-(((2-I)+I*u+Sqrt[3-2*u+u^2])/(((-1-I)+u)*(-I+u)^3))]+(4*I)*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]-(2*I)*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(2*I)*Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4-(3*u^3*(-4-Pi+4*Sqrt[2-2*u+u^2]-4*Sqrt[3-2*u+u^2]+(4*I)*Log[4]-(4*I)*Log[(-1+u)^(-2)]+4*Log[-(((2-I)+I*u+Sqrt[3-2*u+u^2])/(((-1-I)+u)*(-I+u)^3))]+(4*I)*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]-(2*I)*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(2*I)*Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4+(9*u^2*(-4*I-I*Pi+(4*I)*Sqrt[2-2*u+u^2]-(4*I)*Sqrt[3-2*u+u^2]-4*Log[4]+4*Log[(-1+u)^(-2)]+(4*I)*Log[-(((2-I)+I*u+Sqrt[3-2*u+u^2])/(((-1-I)+u)*(-I+u)^3))]-4*Log[(2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2]))/(-1+u)^2]+2*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+2*Log[(2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4)/36+(-4+3*u+6*u^2+(3*(Sqrt[3+(-2+u)*u]+Sqrt[(-1-I)+u]*Sqrt[((-1-I)+u)*((-1+I)+u)^2]*ArcCsch[Sqrt[(-1-I)+u]*Sqrt[(-1+I)+u]]))/2+(15*u*(Sqrt[3+(-2+u)*u]+Sqrt[(-1-I)+u]*Sqrt[((-1-I)+u)*((-1+I)+u)^2]*ArcCsch[Sqrt[(-1-I)+u]*Sqrt[(-1+I)+u]]))/2-(15*u*(Sqrt[2+u^2]+(1+u^2)*ArcCsch[Sqrt[1+u^2]]))/2+6*ArcTan[1-u]-18*u^2*ArcTan[1-u]+6*ArcTan[u]-18*u^2*ArcTan[u]+(15*u*Sqrt[1+u^2]*((2*(1+u^2)*(1-(1+(1+u^2)^(-1))^(3/2)))/3-I*Sqrt[1+u^2]*Beta[-(1+u^2)^(-1),1/2,3/2]))/2-(3*Sqrt[2-2*u+u^2]*((2*(2-2*u+u^2)*(1-(1+(2-2*u+u^2)^(-1))^(3/2)))/3-I*Sqrt[2-2*u+u^2]*Beta[-(2-2*u+u^2)^(-1),1/2,3/2]))/2-(15*u*Sqrt[2-2*u+u^2]*((2*(2-2*u+u^2)*(1-(1+(2-2*u+u^2)^(-1))^(3/2)))/3-I*Sqrt[2-2*u+u^2]*Beta[-(2-2*u+u^2)^(-1),1/2,3/2]))/2+9*u*Log[1+u^2]-3*u^3*Log[1+u^2]-9*u*Log[2-2*u+u^2]+3*u^3*Log[2-2*u+u^2]-6*(-Sqrt[2+(-2+u)*u]+Sqrt[3+(-2+u)*u]+Log[-1+Sqrt[3+(-2+u)*u]])-(9*(-3+Pi-2*u*Sqrt[1+u^2]+2*u*Sqrt[2+u^2]-2*u*Log[1+u^2]+2*Log[-u+Sqrt[1+u^2]]+4*u*Log[-1+Sqrt[2+u^2]]-(2*I)*Log[(-1/2-I/2)*((1-I)+u^2+u*Sqrt[2+u^2])]+(2*I)*Log[(-1/2+I/2)*((1+I)+u^2+u*Sqrt[2+u^2])]))/2+(9*u^2*(4*I-(1+3*I)*Pi+(4*I)*Sqrt[1+u^2]-(4*I)*Sqrt[2+u^2]-I*Log[4]+Log[64]-(4*I)*Log[I/(I+u)^3]+(2*I)*Log[1+u^2]+4*Log[1+u^2+u*Sqrt[1+u^2]]-(4*I)*Log[-1+Sqrt[2+u^2]]-(2-2*I)*Log[(-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-(2+2*I)*Log[(-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4+(3*(-4*I+(1+3*I)*Pi-(4*I)*Sqrt[1+u^2]+(4*I)*Sqrt[2+u^2]+I*Log[4]-Log[64]+(4*I)*Log[I/(I+u)^3]-(2*I)*Log[1+u^2]-4*Log[1+u^2+u*Sqrt[1+u^2]]+(4*I)*Log[-1+Sqrt[2+u^2]]+(2-2*I)*Log[(-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+(2+2*I)*Log[(-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/4+3*(-4+4*Pi+Log[256]-2*(-1+u)*Log[1+u^2]+4*u*Log[-1+Sqrt[2+u^2]]-(2+2*I)*Log[(-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]-(2-2*I)*Log[(2*I)*((1+I)+u^2+u*Sqrt[2+u^2])])+(48*Log[u]+3*u*(-3+u^2)*Log[1+u^2]-2*(5+6*Pi+18*u*Sqrt[1+u^2]-15*u*Sqrt[2+u^2]+6*Log[18]+Log[64]-6*Log[-u+Sqrt[1+u^2]]+12*Log[1+u^2+u*Sqrt[1+u^2]]-9*u*Log[-1+Sqrt[2+u^2]]+3*u^3*Log[-1+Sqrt[2+u^2]]-(6-6*I)*Log[(-3*((1-I)+u^2+u*Sqrt[2+u^2]))/u^2]-(6+6*I)*Log[(-3*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2]))/4+(-17+52*u*Sqrt[1+u^2]-8*u^3*Sqrt[1+u^2]-32*u*Sqrt[2+u^2]+8*u^3*Sqrt[2+u^2]+48*Log[4]+8*Log[8]-96*Log[u]-12*(u+u^3)*Log[1+u^2]-12*Log[-u+Sqrt[1+u^2]]+48*Log[1+u^2+u*Sqrt[1+u^2]]+24*u*Log[-1+Sqrt[2+u^2]]+24*u^3*Log[-1+Sqrt[2+u^2]]-24*Log[(-2+2*I)+(4*I)/u^2-((2-2*I)*Sqrt[2+u^2])/u]-24*Log[((-2-2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2])/16+(9*u*(-4+(3-I)*Pi-4*Sqrt[1+u^2]+4*Sqrt[2+u^2]+Log[4]+(2*I)*Log[8]-(8*I)*Log[u]+4*Log[I/(I+u)^3]-2*Log[1+u^2]+(4*I)*Log[1+u^2+u*Sqrt[1+u^2]]+4*Log[-1+Sqrt[2+u^2]]-(2+2*I)*Log[((-2*I)*((1-I)+u^2+u*Sqrt[2+u^2]))/u^2]+(2-2*I)*Log[((-2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2]))/4-(3*u^3*(-4+(3-I)*Pi-4*Sqrt[1+u^2]+4*Sqrt[2+u^2]+Log[4]+(2*I)*Log[8]-(8*I)*Log[u]+4*Log[I/(I+u)^3]-2*Log[1+u^2]+(4*I)*Log[1+u^2+u*Sqrt[1+u^2]]+4*Log[-1+Sqrt[2+u^2]]-(2+2*I)*Log[((-2*I)*((1-I)+u^2+u*Sqrt[2+u^2]))/u^2]+(2-2*I)*Log[((-2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2]))/4-(3*I)*(-1+(3/4+I/4)*Pi-Sqrt[1+u^2]+Sqrt[2+u^2]+(1-I)*Log[2]+(2*I)*Log[u]+Log[(-I)/(-I+u)^3]-Log[1+u^2]/2-I*Log[1+u^2+u*Sqrt[1+u^2]]+Log[-1+Sqrt[2+u^2]]+(1/2+I/2)*Log[I+(1+I)/u^2+(I*Sqrt[2+u^2])/u]-(1/2-I/2)*Log[((2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2])+9*u*(-1+(3/4+I/4)*Pi-Sqrt[1+u^2]+Sqrt[2+u^2]+(1-I)*Log[2]+(2*I)*Log[u]+Log[(-I)/(-I+u)^3]-Log[1+u^2]/2-I*Log[1+u^2+u*Sqrt[1+u^2]]+Log[-1+Sqrt[2+u^2]]+(1/2+I/2)*Log[I+(1+I)/u^2+(I*Sqrt[2+u^2])/u]-(1/2-I/2)*Log[((2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2])+(9*I)*u^2*(-1+(3/4+I/4)*Pi-Sqrt[1+u^2]+Sqrt[2+u^2]+(1-I)*Log[2]+(2*I)*Log[u]+Log[(-I)/(-I+u)^3]-Log[1+u^2]/2-I*Log[1+u^2+u*Sqrt[1+u^2]]+Log[-1+Sqrt[2+u^2]]+(1/2+I/2)*Log[I+(1+I)/u^2+(I*Sqrt[2+u^2])/u]-(1/2-I/2)*Log[((2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2])-3*u^3*(-1+(3/4+I/4)*Pi-Sqrt[1+u^2]+Sqrt[2+u^2]+(1-I)*Log[2]+(2*I)*Log[u]+Log[(-I)/(-I+u)^3]-Log[1+u^2]/2-I*Log[1+u^2+u*Sqrt[1+u^2]]+Log[-1+Sqrt[2+u^2]]+(1/2+I/2)*Log[I+(1+I)/u^2+(I*Sqrt[2+u^2])/u]-(1/2-I/2)*Log[((2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2])+(9*u^2*(-3+Pi-2*u*Sqrt[1+u^2]+2*u*Sqrt[2+u^2]-2*ArcSinh[u]-2*u*Log[1+u^2]+4*u*Log[-1+Sqrt[2+u^2]]-(2*I)*Log[(-1/2-I/2)*((1-I)+u*(u+Sqrt[2+u^2]))]+(2*I)*Log[(-1/2+I/2)*((1+I)+u*(u+Sqrt[2+u^2]))]))/2-(9*u^2*(-4+4*Pi+Log[256]-2*(-1+u)*Log[1+u^2]+4*u*Log[-1+Sqrt[2+u^2]]-(2+2*I)*Log[(-2*I)*((1-I)+u*(u+Sqrt[2+u^2]))]-(2-2*I)*Log[(2*I)*((1+I)+u*(u+Sqrt[2+u^2]))]))/2+(9*(-3+Pi+2*Sqrt[2-2*u+u^2]-2*u*Sqrt[2-2*u+u^2]-2*Sqrt[3-2*u+u^2]+2*u*Sqrt[3-2*u+u^2]+2*Log[1-u]-2*(-1+u)*Log[2-2*u+u^2]+2*Log[(-1+u-Sqrt[2-2*u+u^2])/(-1+u)]-4*Log[-1+Sqrt[3-2*u+u^2]]+4*u*Log[-1+Sqrt[3-2*u+u^2]]-(2*I)*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(2*I)*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/2-(9*u^2*(-3+Pi+2*Sqrt[2-2*u+u^2]-2*u*Sqrt[2-2*u+u^2]-2*Sqrt[3-2*u+u^2]+2*u*Sqrt[3-2*u+u^2]+2*Log[1-u]-2*(-1+u)*Log[2-2*u+u^2]+2*Log[(-1+u-Sqrt[2-2*u+u^2])/(-1+u)]-4*Log[-1+Sqrt[3-2*u+u^2]]+4*u*Log[-1+Sqrt[3-2*u+u^2]]-(2*I)*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(2*I)*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/2+(3*(4*I-(1+3*I)*Pi+(4*I)*Sqrt[2-2*u+u^2]-(4*I)*Sqrt[3-2*u+u^2]-I*Log[4]+Log[64]-(4*I)*Log[I/(I+u)^3]+(2*I)*Log[2-2*u+u^2]-(4*I)*Log[-1+Sqrt[3-2*u+u^2]]+4*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]-(2-2*I)*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(2+2*I)*Log[(-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4+(9*u^2*(-4*I+(1+3*I)*Pi-(4*I)*Sqrt[2-2*u+u^2]+(4*I)*Sqrt[3-2*u+u^2]+I*Log[4]-Log[64]+(4*I)*Log[I/(I+u)^3]-(2*I)*Log[2-2*u+u^2]+(4*I)*Log[-1+Sqrt[3-2*u+u^2]]-4*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(2-2*I)*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(2+2*I)*Log[(-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/4-3*(-4+4*Pi+Log[4]+Log[64]-4*Log[(-1+u)^(-2)]+4*Log[Sqrt[2-2*u+u^2]/(-1+u)^2]+2*Log[2-2*u+u^2]-2*u*Log[2-2*u+u^2]-4*Log[-1+Sqrt[3-2*u+u^2]]+4*u*Log[-1+Sqrt[3-2*u+u^2]]-(2+2*I)*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(2-2*I)*Log[(2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))])+(9*u^2*(-4+4*Pi+Log[4]+Log[64]-4*Log[(-1+u)^(-2)]+4*Log[Sqrt[2-2*u+u^2]/(-1+u)^2]+2*Log[2-2*u+u^2]-2*u*Log[2-2*u+u^2]-4*Log[-1+Sqrt[3-2*u+u^2]]+4*u*Log[-1+Sqrt[3-2*u+u^2]]-(2+2*I)*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(2-2*I)*Log[(2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/2+(24*Log[(-1+u)^(-2)]-3*(2-3*u^2+u^3)*Log[2-2*u+u^2]+2*(5+6*Pi-18*Sqrt[2-2*u+u^2]+18*u*Sqrt[2-2*u+u^2]+15*Sqrt[3-2*u+u^2]-15*u*Sqrt[3-2*u+u^2]+6*Log[18]+Log[64]-6*Log[1-u+Sqrt[2-2*u+u^2]]+3*(2-3*u^2+u^3)*Log[-1+Sqrt[3-2*u+u^2]]+12*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]-(6-6*I)*Log[(-3*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]-(6+6*I)*Log[(-3*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]))/4+(17+44*Sqrt[2-2*u+u^2]-28*u*Sqrt[2-2*u+u^2]-24*u^2*Sqrt[2-2*u+u^2]+8*u^3*Sqrt[2-2*u+u^2]-24*Sqrt[3-2*u+u^2]+8*u*Sqrt[3-2*u+u^2]+24*u^2*Sqrt[3-2*u+u^2]-8*u^3*Sqrt[3-2*u+u^2]-8*Log[8]-2*Log[281474976710656]-48*Log[(-1+u)^(-2)]+12*(-2+4*u-3*u^2+u^3)*Log[2-2*u+u^2]+12*Log[1-u+Sqrt[2-2*u+u^2]]+48*Log[-1+Sqrt[3-2*u+u^2]]-96*u*Log[-1+Sqrt[3-2*u+u^2]]+72*u^2*Log[-1+Sqrt[3-2*u+u^2]]-24*u^3*Log[-1+Sqrt[3-2*u+u^2]]-48*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+24*Log[((-2+6*I)-(2-2*I)*u^2+(2-2*I)*Sqrt[3-2*u+u^2]-(2-2*I)*u*(-2+Sqrt[3-2*u+u^2]))/(-1+u)^2]+24*Log[((-2-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2])/16-(9*u*(-4+(3-I)*Pi-4*Sqrt[2-2*u+u^2]+4*Sqrt[3-2*u+u^2]+Log[4]+(2*I)*Log[8]+(4*I)*Log[(-1+u)^(-2)]+4*Log[I/(I+u)^3]-2*Log[2-2*u+u^2]+4*Log[-1+Sqrt[3-2*u+u^2]]+(4*I)*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]-(2+2*I)*Log[((-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]+(2-2*I)*Log[((-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]))/4+(3*u^3*(-4+(3-I)*Pi-4*Sqrt[2-2*u+u^2]+4*Sqrt[3-2*u+u^2]+Log[4]+(2*I)*Log[8]+(4*I)*Log[(-1+u)^(-2)]+4*Log[I/(I+u)^3]-2*Log[2-2*u+u^2]+4*Log[-1+Sqrt[3-2*u+u^2]]+(4*I)*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]-(2+2*I)*Log[((-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]+(2-2*I)*Log[((-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]))/4+(3*I)*(-1+(3/4+I/4)*Pi-Sqrt[2-2*u+u^2]+Sqrt[3-2*u+u^2]+(1-I)*Log[2]-I*Log[(-1+u)^(-2)]+Log[(-I)/(-I+u)^3]-Log[2-2*u+u^2]/2+Log[-1+Sqrt[3-2*u+u^2]]-I*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(1/2+I/2)*Log[((1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2]))/(-1+u)^2]-(1/2-I/2)*Log[((2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2])-9*u*(-1+(3/4+I/4)*Pi-Sqrt[2-2*u+u^2]+Sqrt[3-2*u+u^2]+(1-I)*Log[2]-I*Log[(-1+u)^(-2)]+Log[(-I)/(-I+u)^3]-Log[2-2*u+u^2]/2+Log[-1+Sqrt[3-2*u+u^2]]-I*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(1/2+I/2)*Log[((1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2]))/(-1+u)^2]-(1/2-I/2)*Log[((2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2])-(9*I)*u^2*(-1+(3/4+I/4)*Pi-Sqrt[2-2*u+u^2]+Sqrt[3-2*u+u^2]+(1-I)*Log[2]-I*Log[(-1+u)^(-2)]+Log[(-I)/(-I+u)^3]-Log[2-2*u+u^2]/2+Log[-1+Sqrt[3-2*u+u^2]]-I*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(1/2+I/2)*Log[((1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2]))/(-1+u)^2]-(1/2-I/2)*Log[((2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2])+3*u^3*(-1+(3/4+I/4)*Pi-Sqrt[2-2*u+u^2]+Sqrt[3-2*u+u^2]+(1-I)*Log[2]-I*Log[(-1+u)^(-2)]+Log[(-I)/(-I+u)^3]-Log[2-2*u+u^2]/2+Log[-1+Sqrt[3-2*u+u^2]]-I*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(1/2+I/2)*Log[((1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2]))/(-1+u)^2]-(1/2-I/2)*Log[((2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]))/36+(u*(-1-2*u-Sqrt[3+(-2+u)*u]+Sqrt[1+u^2]-Sqrt[2-2*u+u^2]+Sqrt[3-2*u+u^2]-Sqrt[(-1-I)+u]*Sqrt[((-1-I)+u)*((-1+I)+u)^2]*ArcCsch[Sqrt[(-1-I)+u]*Sqrt[(-1+I)+u]]+(1+u^2)*ArcCsch[Sqrt[1+u^2]]+(-2*Sqrt[1+u^2]+Sqrt[2+u^2]-2*u^2*(Sqrt[1+u^2]-Sqrt[2+u^2])-3*(1+u^2)*ArcCsch[Sqrt[1+u^2]])/3+4*u*ArcTan[1-u]+4*u*ArcTan[u]+Sqrt[2-2*u+u^2]*((2*(2-2*u+u^2)*(1-(1+(2-2*u+u^2)^(-1))^(3/2)))/3-I*Sqrt[2-2*u+u^2]*Beta[-(2-2*u+u^2)^(-1),1/2,3/2])-I*Log[(-1+u)^(-2)]-(2*I)*Log[u]-Log[1+u^2]/2+u^2*Log[1+u^2]+Log[2-2*u+u^2]/2-u^2*Log[2-2*u+u^2]+2*(-Sqrt[2+(-2+u)*u]+Sqrt[3+(-2+u)*u]+Log[-1+Sqrt[3+(-2+u)*u]])+I*Log[1+u^2+u*Sqrt[1+u^2]]-Log[-1+Sqrt[2+u^2]]-(1/2+I/2)*Log[I+(1+I)/u^2+(I*Sqrt[2+u^2])/u]+(u*(-4*I+(1+3*I)*Pi-(4*I)*Sqrt[1+u^2]+(4*I)*Sqrt[2+u^2]+I*Log[4]-Log[64]+(4*I)*Log[I/(I+u)^2]-(2*I)*Log[1+u^2]-4*Log[1+u^2+u*Sqrt[1+u^2]]+(4*I)*Log[-1+Sqrt[2+u^2]]+(2-2*I)*Log[(-2*I)*((1-I)+u^2+u*Sqrt[2+u^2])]+(2+2*I)*Log[(-2*I)*((1+I)+u^2+u*Sqrt[2+u^2])]))/2+(4-(3-I)*Pi+4*Sqrt[1+u^2]-4*Sqrt[2+u^2]-Log[4]-(2*I)*Log[8]+(8*I)*Log[u]-4*Log[I/(I+u)^2]+2*Log[1+u^2]-(4*I)*Log[1+u^2+u*Sqrt[1+u^2]]-4*Log[-1+Sqrt[2+u^2]]+(2+2*I)*Log[((-2*I)*((1-I)+u^2+u*Sqrt[2+u^2]))/u^2]-(2-2*I)*Log[((-2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2])/4+(u^2*(-4+(3-I)*Pi-4*Sqrt[1+u^2]+4*Sqrt[2+u^2]+Log[4]+(2*I)*Log[8]-(8*I)*Log[u]+4*Log[I/(I+u)^2]-2*Log[1+u^2]+(4*I)*Log[1+u^2+u*Sqrt[1+u^2]]+4*Log[-1+Sqrt[2+u^2]]-(2+2*I)*Log[((-2*I)*((1-I)+u^2+u*Sqrt[2+u^2]))/u^2]+(2-2*I)*Log[((-2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2]))/4-(2*I)*u*(-1+(3/4+I/4)*Pi-Sqrt[1+u^2]+Sqrt[2+u^2]+(1-I)*Log[2]+(2*I)*Log[u]+Log[(-I)/(-I+u)^2]-Log[1+u^2]/2-I*Log[1+u^2+u*Sqrt[1+u^2]]+Log[-1+Sqrt[2+u^2]]+(1/2+I/2)*Log[I+(1+I)/u^2+(I*Sqrt[2+u^2])/u]-(1/2-I/2)*Log[((2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2])+u^2*(-1+(3/4+I/4)*Pi-Sqrt[1+u^2]+Sqrt[2+u^2]+(1-I)*Log[2]+(2*I)*Log[u]+Log[(-I)/(-I+u)^2]-Log[1+u^2]/2-I*Log[1+u^2+u*Sqrt[1+u^2]]+Log[-1+Sqrt[2+u^2]]+(1/2+I/2)*Log[I+(1+I)/u^2+(I*Sqrt[2+u^2])/u]-(1/2-I/2)*Log[((2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2])+(1/2-I/2)*Log[((2*I)*((1+I)+u^2+u*Sqrt[2+u^2]))/u^2]+Log[-1+Sqrt[3-2*u+u^2]]+u*(3-Pi+2*u*Sqrt[1+u^2]-2*u*Sqrt[2+u^2]+2*ArcSinh[u]+2*u*Log[1+u^2]-4*u*Log[-1+Sqrt[2+u^2]]+(2*I)*Log[(-1/2-I/2)*((1-I)+u*(u+Sqrt[2+u^2]))]-(2*I)*Log[(-1/2+I/2)*((1+I)+u*(u+Sqrt[2+u^2]))])+u*(-4+4*Pi+Log[256]-2*(-1+u)*Log[1+u^2]+4*u*Log[-1+Sqrt[2+u^2]]-(2+2*I)*Log[(-2*I)*((1-I)+u*(u+Sqrt[2+u^2]))]-(2-2*I)*Log[(2*I)*((1+I)+u*(u+Sqrt[2+u^2]))])-I*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(1/2+I/2)*Log[((1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2]))/(-1+u)^2]+u*(-3+Pi+2*Sqrt[2-2*u+u^2]-2*u*Sqrt[2-2*u+u^2]-2*Sqrt[3-2*u+u^2]+2*u*Sqrt[3-2*u+u^2]+2*Log[1-u]-2*(-1+u)*Log[2-2*u+u^2]+2*Log[(-1+u-Sqrt[2-2*u+u^2])/(-1+u)]-4*Log[-1+Sqrt[3-2*u+u^2]]+4*u*Log[-1+Sqrt[3-2*u+u^2]]-(2*I)*Log[(-2-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]+(2*I)*Log[(-2+2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))])+(u*(4*I-(1+3*I)*Pi+(4*I)*Sqrt[2-2*u+u^2]-(4*I)*Sqrt[3-2*u+u^2]-I*Log[4]+Log[64]-(4*I)*Log[I/(I+u)^2]+(2*I)*Log[2-2*u+u^2]-(4*I)*Log[-1+Sqrt[3-2*u+u^2]]+4*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]-(2-2*I)*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(2+2*I)*Log[(-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]))/2-u*(-4+4*Pi+Log[4]+Log[64]-4*Log[(-1+u)^(-2)]+4*Log[Sqrt[2-2*u+u^2]/(-1+u)^2]+2*Log[2-2*u+u^2]-2*u*Log[2-2*u+u^2]-4*Log[-1+Sqrt[3-2*u+u^2]]+4*u*Log[-1+Sqrt[3-2*u+u^2]]-(2+2*I)*Log[(-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))]-(2-2*I)*Log[(2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2]))])+(-4+(3-I)*Pi-4*Sqrt[2-2*u+u^2]+4*Sqrt[3-2*u+u^2]+Log[4]+(2*I)*Log[8]+(4*I)*Log[(-1+u)^(-2)]+4*Log[I/(I+u)^2]-2*Log[2-2*u+u^2]+4*Log[-1+Sqrt[3-2*u+u^2]]+(4*I)*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]-(2+2*I)*Log[((-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]+(2-2*I)*Log[((-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2])/4-(u^2*(-4+(3-I)*Pi-4*Sqrt[2-2*u+u^2]+4*Sqrt[3-2*u+u^2]+Log[4]+(2*I)*Log[8]+(4*I)*Log[(-1+u)^(-2)]+4*Log[I/(I+u)^2]-2*Log[2-2*u+u^2]+4*Log[-1+Sqrt[3-2*u+u^2]]+(4*I)*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]-(2+2*I)*Log[((-2*I)*((2-I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]+(2-2*I)*Log[((-2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]))/4+(2*I)*u*(-1+(3/4+I/4)*Pi-Sqrt[2-2*u+u^2]+Sqrt[3-2*u+u^2]+(1-I)*Log[2]-I*Log[(-1+u)^(-2)]+Log[(-I)/(-I+u)^2]-Log[2-2*u+u^2]/2+Log[-1+Sqrt[3-2*u+u^2]]-I*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(1/2+I/2)*Log[((1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2]))/(-1+u)^2]-(1/2-I/2)*Log[((2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2])-u^2*(-1+(3/4+I/4)*Pi-Sqrt[2-2*u+u^2]+Sqrt[3-2*u+u^2]+(1-I)*Log[2]-I*Log[(-1+u)^(-2)]+Log[(-I)/(-I+u)^2]-Log[2-2*u+u^2]/2+Log[-1+Sqrt[3-2*u+u^2]]-I*Log[2+u^2-Sqrt[2-2*u+u^2]+u*(-2+Sqrt[2-2*u+u^2])]+(1/2+I/2)*Log[((1+2*I)+I*u^2-I*Sqrt[3-2*u+u^2]+I*u*(-2+Sqrt[3-2*u+u^2]))/(-1+u)^2]-(1/2-I/2)*Log[((2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2])-(1/2-I/2)*Log[((2*I)*((2+I)+u^2-Sqrt[3-2*u+u^2]+u*(-2+Sqrt[3-2*u+u^2])))/(-1+u)^2]))/4;

Plot[Evaluate[i2],{u,0,1},PlotStyle->Red]

...wrong...

NIntegrate[i2,{u,0,1},MaxRecursion->20,Workin]

GenerateConditions->True

v5.0

Timing[i2=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1},GenerateConditions->True]]

N[%]

...wrong...

v5.1

Timing[i2=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1},GenerateConditions->True]]

N[%]

...wrong...

v5.2

Timing[i2=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1},GenerateConditions->True]]

N[i2,20]

...wrong...

v6

Timing[i2=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1},GenerateConditions->True]]

N[%]

...wrong...

Propagating Assumptions

v5.0.1

(i2=(1/5)IntegrateWithAssumptions[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1}])//Timing

N[%]

...wrong...

v5.1

Timing[i2=(1/5)IntegrateWithAssumptions[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1}]]

N[%]

...wrong...

v6

Timing[i2=(1/5)IntegrateWithAssumptions[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1}]]

N[%]

...wrong...

GenerateConditions->False for 1st two and Assumptions for 3rd

v5.0.1

Timing[first=4/5Integrate[Sqrt[x^2+y^2+(z-w)^2],{x,0,1},{y,0,1},{w,0,1},{z,0,1}]]

first=(1/900) (168 (Sqrt[2])-24 (Sqrt[3])-44 π+720 ArcCsch[(Sqrt[2])]+72 ArcSinh[1]-486 ArcSinh[(1/(Sqrt[2]))]+33 Log[2]-10 Log[64]-240 Log[-1+(Sqrt[3])]+294 Log[1+(Sqrt[3])])

Timing[second=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{z,0,1},{y,0,1},{w,0,1},{u,0,1}]]

Timing[ii=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{y,0,1},{u,0,1},GenerateConditions->False]]

Timing[i2=Integrate[ii,{w,0,1},Assumptions->0<z<1]]

NIntegrate[i2,{z,0,1},WorkingPrecision->20]

Timing[second=Integrate[i2,{z,0,1}]]

v5.1

Timing[ii=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{y,0,1},{u,0,1},GenerateConditions->False]]

Timing[i2=Integrate[ii,{w,0,1},Assumptions->0<z<1]]

NIntegrate[i2,{z,0,1},WorkingPrecision->20]

Timing[second=Integrate[i2,{z,0,1}]]

v6.0

Timing[ii=(1/5)Integrate[Sqrt[1+(y-u)^2+(z-w)^2],{y,0,1},{u,0,1},GenerateConditions->False]]

Timing[i2=Integrate[ii,{w,0,1},Assumptions->0<z<1]]

NIntegrate[i2,{z,0,1},WorkingPrecision->20]

Timing[second=Integrate[i2,{z,0,1}]]

Combined Integrals

N[first+second,100]

expr=Simplify[TrigToExp[FullSimplify[first+second]]]

expr2=expr/.n_Integer Log[x_]:>log[x^n]

expr2=(1/150) (8+34 (Sqrt[2])-12 (Sqrt[3])-14 π+log[(1/524288)]+log[(((1+(Sqrt[2])))^32)]+log[(((3+2 (Sqrt[2])))^5)]+log[(1/(((-1+(Sqrt[3])))^70))]+log[(((1+(Sqrt[3])))^108)]+log[(1/(((2+(Sqrt[3])))^5))])

FullSimplify[ExpandAll[Times@@First/@Cases[expr2,_log,{1,∞}]]]

Log[1+Sqrt[2],5964153172084899+4217293152016490 (Sqrt[2])]//N

Log[2+Sqrt[3],(552794861161443374798157973077388454680145589601+319156261898201133032451783177661812885799840920 (Sqrt[3]))]//N

(1/150) (8+34 (Sqrt[2])-12 (Sqrt[3])-14 π+42Log[1+Sqrt[2]]+84Log[2+Sqrt[3]])

(1/150) (8+34 (Sqrt[2])-12 (Sqrt[3])-14 π+42Log[1+Sqrt[2]]+84Log[2+Sqrt[3]])//Simplify

N[%,100]

Points on Different Faces

Trott's Laplace Transform Method

Note that the following integral is equivalent to (Sqrt[s])

Integrate[-1/Sqrt[Pi]/2 p^(-3/2) (Exp[-p s]-1),{p,0,Infinity},Assumptions->Re[s]>0]

so the square root (which is what causes most of the difficulty in symbolic evaluation of the multiple integrals) can be eliminated at the
expense of introducing one additional integral.

A second trick involves rewriting the integrands using partial integration to place them in a form that Mathematica can do out of the box.

partialIntegrate[p_Plus]:=partialIntegrate/@p;
partialIntegrate[a_?NumericQ b_]:=a partialIntegrate[b]

partialIntegrateRule=partialIntegrate[a_ z^n_?(#<-1&)]:>-partialIntegrate[(1/(n+1) z^(n+1)) D[a,z]];

dropSingularTerms[expr_,z_]:=expr-Normal[Series[expr,{z,0,-1}]]

Part 1

expr1=4/5(Sqrt[x^2+y^2+(z-w)^2])/.Sqrt[x_]:>(Exp[-p x]-1)

int1=-((p^(-3/2))/(Sqrt[π]))Integrate[expr1 ,{x,0,1},{y,0,1},{z,0,1},{w,0,1}]

Plot[int1,{p,0,10},PlotStyle->Red]

int1/.p->100.

(part1=NIntegrate[int1,{p,0,10^6},PrecisionGoal->5,WorkingPrecision->30,MaxRecursion->10])//Timing

Last[%]/2

Value

0.6966214608239568`

Direct

Integrate[int1,{p,0,∞}]//Timing

Transformed

int1z=FullSimplify[int1 z/.p->z^2,z>0]

NIntegrate[int1z,{z,0,10^3},PrecisionGoal->4,WorkingPrecision->10,MaxRecursion->10]//Timing

Integrate[int1z,{z,0,∞}]//Timing

DropSingularTerms

int1zds=dropSingularTerms[#,z]&@(FixedPoint[ExpandAll[#/.partialIntegrateRule]&,partialIntegrate[ExpandAll[int1z]]]/.partialIntegrate->Identity)

Plot[{int1z,int1zds},{z,0.01,10},PlotStyle->List/@{Red,Blue},PlotRange->All]

NIntegrate[int1zds,{z,0,10^3},PrecisionGoal->4,WorkingPrecision->10,MaxRecursion->10]//Timing

Integrate[int1zds,{z,0,∞}]

N[%,20]

Part 2

Direct

expr2=1/5(Sqrt[(((y-u))^2)+(((z-w))^2)+1])/.Sqrt[x_]:>(Exp[-p x]-1)

int2=FullSimplify[-((p^(-3/2))/(Sqrt[π]))Integrate[expr2,{u,0,1},{w,0,1},{y,0,1},{z,0,1}]]

Plot[int2,{p,0,10},PlotStyle->Red]

int2/.p->100.

(part2=NIntegrate[int2,{p,0,10^6},PrecisionGoal->5,WorkingPrecision->30,MaxRecursion->10])//Timing

Integrate[int2,{p,0,∞}]//Timing

Value

N[(1/450) (24+18 (Sqrt[2])-24 (Sqrt[3])-20 π-60 ArcCsch[(Sqrt[2])]-60 ArcSinh[1]+45 Log[2]+75 Log[3+2 (Sqrt[2])]-90 Log[-1+(Sqrt[3])]-75 Log[2+(Sqrt[3])]+10 Log[3650401+2107560 (Sqrt[3])]),20]

Transformed

int2z=FullSimplify[int2 z/.p->z^2,z>0]

NIntegrate[int2z,{z,0,10^3},PrecisionGoal->4,WorkingPrecision->10,MaxRecursion->10]//Timing

Integrate[int2z,{z,0,∞}]//Timing

ExpandAll[int2z].

DropSingularTerms

int2zds=dropSingularTerms[#,z]&@(FixedPoint[ExpandAll[#/.partialIntegrateRule]&,partialIntegrate[ExpandAll[int2z]]]/.partialIntegrate->Identity)

Plot[{int2z,int2zds},{z,0.01,10},PlotStyle->List/@{Red,Blue},PlotRange->All]

NIntegrate[int2zds,{z,0,10^3},PrecisionGoal->4,WorkingPrecision->10,MaxRecursion->10]//Timing

Integrate[int2zds,{z,0,∞}]

N[%,20]

Both Parts

(part1+part2)/2

4/75+(17*Sqrt[2])/75-(2*Sqrt[3])/25-(7*Pi)/75+(28*Log[Sqrt[3/2]+1/Sqrt[2]])/25+(7*Log[1+Sqrt[2]])/25//N

(expr=4/5 Integrate[(Exp[-p (x^2+y^2+(z-w)^2)]-1),{x,0,1},{y,0,1},{z,0,1},{w,0,1}]+1/5 Integrate[(Exp[-p (1+(y-u)^2+(z-w)^2)]-1),{u,0,1},{w,0,1},{y,0,1},{z,0,1}])//Timing

intz=2z PowerExpand[-1/Sqrt[Pi]/2 p^(-3/2) expr/.p->z^2]

integrand0=dropSingularTerms[#,z]&@(FixedPoint[ExpandAll[#/.partialIntegrateRule]&,partialIntegrate[ExpandAll[intz]]]/.partialIntegrate->Identity)

Integrate[integrand0,{z,0,Infinity}]//Timing

integrand=integrand0//FullSimplify

Integrate[integrand,{z,0,Infinity}]//TrigToExp//Timing

N[Last[%13],20]

Sums

First Sum

N[((Sqrt[2π])/5)Sum[((Hypergeometric2F1[1/2,-n+2,3/2,1/2])/((2n+1)Gamma[n+2]Gamma[5/2-n])),{n,2,500}]+(4/15)(Sqrt[2])+(2/5)Log[(Sqrt[2])+1]-(1/75)π,10]//Timing

v4.2

((Sqrt[2π])/5)NSum[((Hypergeometric2F1[1/2,-n+2,3/2,1/2])/((2n+1)Gamma[n+1]Gamma[5/2-n])),{n,2,∞}]+(4/15)(Sqrt[2])+(2/5)Log[(Sqrt[2])+1]-(1/75)π

v5

Hypergeometric2F1[1/2,-n+2,3/2,1/2]

Sum[((Hypergeometric2F1[1/2,-n+2,3/2,1/2])/((2n+1)Gamma[n+2]Gamma[5/2-n])),{n,2,∞}]//Timing

((Sqrt[2π])/5)NSum[((Hypergeometric2F1[1/2,-n+2,3/2,1/2])/((2n+1)Gamma[n+1]Gamma[5/2-n])),{n,2,∞},Method->Fit]+(4/15)(Sqrt[2])+(2/5)Log[(Sqrt[2])+1]-(1/75)π//Timing

NSum[((Hypergeometric2F1[1/2,-n+2,3/2,1/2])/((2n+1)Gamma[n+2]Gamma[5/2-n])),{n,2,∞}]//Timing

...hangs...

v6

((Sqrt[2π])/5)NSum[((Hypergeometric2F1[1/2,-n+2,3/2,1/2])/((2n+1)Gamma[n+1]Gamma[5/2-n])),{n,2,∞}]+(4/15)(Sqrt[2])+(2/5)Log[(Sqrt[2])+1]-(1/75)π//Timing

...hangs...

((Sqrt[2π])/5)NSum[((Hypergeometric2F1[1/2,-n+2,3/2,1/2])/((2n+1)Gamma[n+1]Gamma[5/2-n])),{n,2,∞},Method->Fit]+(4/15)(Sqrt[2])+(2/5)Log[(Sqrt[2])+1]-(1/75)π//Timing

Second Sum

This sum is classically divergent

N[exact=(1/450) (24+18 (Sqrt[2])-24 (Sqrt[3])-20 π-60 ArcCsch[(Sqrt[2])]-60 ArcSinh[1]+45 Log[2]+75 Log[3+2 (Sqrt[2])]-90 Log[-1+(Sqrt[3])]-75 Log[2+(Sqrt[3])]+10 Log[3650401+2107560 (Sqrt[3])]),10]

N[((Sqrt[π])/10)Sum[(1/((2n+1)Gamma[n+2]Gamma[3/2-n]))HypergeometricPFQ[{1,1/2,-1/2-n,-n-1},{2,1/2-n,3/2},-1],{n,0,10}]-(2/25)+((Sqrt[2])/50)+(1/10)Log[(Sqrt[2])+1],10]//Timing

((Sqrt[π])/10)NSum[(1/((2n+1)Gamma[n+2]Gamma[3/2-n]))HypergeometricPFQ[{1,1/2,-1/2-n,-n-1},{2,1/2-n,3/2},-1],{n,0,∞}]-(2/25)+((Sqrt[2])/50)+(1/10)Log[(Sqrt[2])+1]//Timing

Look at Terms

N[Table[(1/((2n+1)Gamma[n+2]Gamma[3/2-n]))HypergeometricPFQ[{1,1/2,-1/2-n,-n-1},{2,1/2-n,3/2},-1],{n,100}],10]

ListPlot[Table[(1/((2n+1)Gamma[n+2]Gamma[3/2-n]))HypergeometricPFQ[{1,1/2,-1/2-n,-n-1},{2,1/2-n,3/2},-1],{n,20}],PlotJoined->True,PlotStyle->Red,PlotRange->All]

(Sqrt[Pi]/10)*NSum[HypergeometricPFQ[{1,1/2,-2^(-1)-n,-n-1},{2,1/2-n,3/2},-1]/((2*n+1)*Gamma[n+2]*Gamma[3/2-n]),{n,0,Infinity},Method->Fit,WorkingPrecision->100,NSumTerms->100,NSumExtraTerms->100]-2/25+Sqrt[2]/50+(1/10)*Log[Sqrt[2]+1]

((Sqrt[π])/10)NSum[(1/((2n+1)Gamma[n+2]Gamma[3/2-n]))HypergeometricPFQ[{1,1/2,-1/2-n,-n-1},{2,1/2-n,3/2},-1],{n,0,∞}]-(2/25)+((Sqrt[2])/50)+(1/10)Log[(Sqrt[2])+1]//Timing

nsum=((Sqrt[π])/10)NSum[HypergeometricPFQ[{1,1/2,-2^(-1)-n,-n-1},{2,1/2-n,3/2},-1]/((2*n+1)Gamma[n+2]Gamma[3/2-n]),{n,0,∞},Method->Fit,WorkingPrecision->100]-(2/25)+((Sqrt[2])/50)+(1/10)Log[(Sqrt[2])+1]//Timing

nsum-exact

(nsum=((Sqrt[π])/10)NSum[HypergeometricPFQ[{1,1/2,-2^(-1)-n,-n-1},{2,1/2-n,3/2},-1]/((2*n+1)Gamma[n+2]Gamma[3/2-n]),{n,0,∞},Method->Fit,WorkingPrecision->100,NSumTerms->100,NSumExtraTerms->100]-(2/25)+((Sqrt[2])/50)+(1/10)Log[(Sqrt[2])+1])//Timing

N[exact,40]

nsum-exact

(nsum=((Sqrt[π])/10)NSum[HypergeometricPFQ[{1,1/2,-2^(-1)-n,-n-1},{2,1/2-n,3/2},-1]/((2*n+1)Gamma[n+2]Gamma[3/2-n]),{n,0,∞},Method->Fit,WorkingPrecision->100,NSumTerms->90,NSumExtraTerms->100]-(2/25)+((Sqrt[2])/50)+(1/10)Log[(Sqrt[2])+1])//Timing

nsum-exact

values=Table[((Sqrt[π])/10)NSum[HypergeometricPFQ[{1,1/2,-2^(-1)-n,-n-1},{2,1/2-n,3/2},-1]/((2*n+1)Gamma[n+2]Gamma[3/2-n]),{n,0,∞},Method->Fit,WorkingPrecision->100,NSumTerms->t1,NSumExtraTerms->t2]-(2/25)+((Sqrt[2])/50)+(1/10)Log[(Sqrt[2])+1],{t1,10,150,10},{t2,10,150,10}];//Timing

values-exact//MatrixForm

ListDensityPlot[-Log[10,Abs[values-exact]]/.Indeterminate->20,ColorFunction->(Hue[#/2]&)]

Max[Map[Precision,Abs[exact-values],{2}]]

ListDensityPlot[Map[Precision,Abs[values-exact],{2}]/.Indeterminate->20,ColorFunction->(Hue[#/2]&)]

(nsum=((Sqrt[π])/10)NSum[HypergeometricPFQ[{1,1/2,-2^(-1)-n,-n-1},{2,1/2-n,3/2},-1]/((2*n+1)Gamma[n+2]Gamma[3/2-n]),{n,0,∞},Method->Fit,WorkingPrecision->100,NSumTerms->50,NSumExtraTerms->100]-(2/25)+((Sqrt[2])/50)+(1/10)Log[(Sqrt[2])+1])//Timing

nsum-exact

NSum[(1/((2*n+1)*Gamma[n+2]*Gamma[3/2-n]))*HypergeometricPFQ[{1,1/2,-2^(-1)-n,-n-1},{2,1/2-n,3/2},-1],{n,0,∞}]//Timing

Term by Term

(1/((2*n+1)*Gamma[n+2]*Gamma[3/2-n]))*HypergeometricPFQ[{1,1/2,-2^(-1)-n,-n-1},{2,1/2-n,3/2},-1]/.n->0

Relationship between Δ(3) and (Subscript[Δ, f])(3)

N[deltaf=(1/210) (8+34 (Sqrt[2])-12 (Sqrt[3])-14 π+42Log[1+ (Sqrt[2])]+84Log[ 2+(Sqrt[3])])]

N[deltaf3=(4/75)+((17 (Sqrt[2]))/75)-((2 (Sqrt[3]))/25)-((7 π)/75)+(28/25) Log[(Sqrt[(3/2)])+(1/(Sqrt[2]))]+(7/25) Log[1+(Sqrt[2])]]

FullSimplify[deltaf/deltaf3]