Supersphere

Author

Eric W. Weisstein
July 31, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/Supersphere.nb.

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Figure

Row[Table[
ContourPlot3D[Abs[x]^n+Abs[y]^n+Abs[z]^n==1,
{x,-1,1},{y,-1,1},{z,-1,1},
Boxed->False,Axes->False,Mesh->False,Method->{"ShrinkWrap"->True},PlotLabel->TraditionalForm[HoldForm[n]==n],LabelStyle->Directive[Black,FontFamily->"Times"]],
{n,4}],Spacer[5],ImageSize->500,Alignment->Center
]

GraphicsRow[Region[#,PerformanceGoal->"Quality"]&/@Table[ImplicitRegion[((Abs[x])^n)+((Abs[y])^n)+((Abs[z])^n)<=1,{x,y,z}],{n,{1,2,3,4}}]]

Centroid

RegionCentroid[ImplicitRegion[((Abs[x])^n)+((Abs[y])^n)+((Abs[z])^n)<=1,{x,y,z}]]//Timing

Semialgebraic equation

TextGrid[Table[{n,Factor/@GroebnerBasis[{u+v+w==1,(x^n)==(u^n),(y^n)==(v^n),(z^n)==(w^n)},{x,y,z},{u,v,w}]},{n,3}],Dividers->All]

Surface area

Numerics GuideBook, §1.8, p. 301

Trott, M. GuideBook for Numerics, p. 301.

(∫_0^(π/4))(∫_0^((sec^(-1))((Sqrt[(tan^2)(φ)+2]))))(Sqrt[(((sin^(2 n))(ϑ) ((sin^(2 n-2))(φ)+(cos^(2 n-2))(φ))+(sin^2)(ϑ) (cos^(2 n-2))(ϑ))/((((sin^n)(ϑ) ((sin^n)(φ)+(cos^n)(φ))+(cos^n)(ϑ)))^((2 (n+2))/n)))])\[DifferentialD]ϑ\[DifferentialD]φ

SupersphereAreaTrott[n_, opts___] := 48 NIntegrate[
 (* integrand comes from:
   Module[{r, e, f, g, dA},
   (* radius vector in spherical coordinates *)
    r =  1/(Cos[ϑ]^n + Cos[φ]^n Sin[ϑ]^n + Sin[ϑ]^n Sin[φ]^n)^(1/n)*
                              {Cos[φ] Sin[ϑ], Sin[φ] Sin[ϑ], Cos[ϑ]};
   (* surface area element *)
    e = D[r, φ].D[r, φ]; f = D[r, φ].D[r, ϑ]; g = D[r, ϑ].D[r, ϑ];
    dA = Sqrt[e g - f^2] // Simplify] *)
  Sqrt[(Cos[ϑ]^(2n - 2) Sin[ϑ]^2 + Sin[ϑ]^(2n)*
       (Cos[φ]^(2n - 2) + Sin[φ]^(2n - 2)))/
       (Cos[ϑ]^n + Sin[ϑ]^n (Cos[φ]^n + Sin[φ]^n))^((2(2 + n))/n)], 
   (* integrate over 1/48th of the solid angle *)
   {φ, 0, Pi/4}, {ϑ, 0, ArcCos[1/Sqrt[2 + Tan[φ]^2]]}, opts,
   (* option values appropriate for a plot *)
   PrecisionGoal -> 4, AccuracyGoal -> 6, Method -> GaussKronrod]

Baillie

Robert Baillie.  "Superspheres."  January 29, 2022.

SupersphereAreaBaillie[k_]:=Module[{
a=(3^(-1/k)),
b=(2^(-1/k)),
i1,i2
},
i1=NIntegrate[(Sqrt[1+(((x^(2k-2))+(y^(2k-2)))/(((1-(x^k)-(y^k)))^((2k-2)/k)))]),{x,0,a},{y,0,x}];
i2=NIntegrate[(Sqrt[1+(((x^(2k-2))+(y^(2k-2)))/(((1-(x^k)-(y^k)))^((2k-2)/k)))]),{x,a,b},{y,0,(((1-2(x^k)))^(1/k))}];
48(i1+i2)
]

1 == octahedron

SurfaceArea[ImplicitRegion[Abs[x]+Abs[y]+Abs[z]<=1,{x,y,z}]]

SurfaceArea[ImplicitRegion[Abs[x]+Abs[y]+Abs[z]<=1,{x,y,z}],PerformanceGoal->"Quality"]//Timing

4(Sqrt[3])//N

SupersphereAreaBaillie[1]

SupersphereAreaTrott[1]

2 == sphere

SurfaceArea

SurfaceArea[ImplicitRegion[((Abs[x])^2)+((Abs[y])^2)+((Abs[z])^2)<=1,{x,y,z}]]

4Pi//N

SupersphereAreaBaillie[2]

SupersphereAreaTrott[2]

With[{n=2},48(∫_0^(π/4))(∫_0^(ArcCos[(1/(Sqrt[2+((Tan[φ])^2)]))]))(Sqrt[((((Cos[ϑ])^(2 n-2)) ((Sin[ϑ])^2)+((Sin[ϑ])^(2 n)) (((Cos[φ])^(2 n-2))+((Sin[φ])^(2 n-2))))/(((((Cos[ϑ])^n)+((Sin[ϑ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^(2 (n+2)/n)))])\[DifferentialD]ϑ\[DifferentialD]φ]//Timing

With[{n=2},48(∫_0^(π/4))(∫_0^(ArcSec[(Sqrt[2+((Tan[φ])^2)])]))(Sqrt[((((Cos[ϑ])^(2 n-2)) ((Sin[ϑ])^2)+((Sin[ϑ])^(2 n)) (((Cos[φ])^(2 n-2))+((Sin[φ])^(2 n-2))))/(((((Cos[ϑ])^n)+((Sin[ϑ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^(2 (n+2)/n)))])\[DifferentialD]ϑ\[DifferentialD]φ]//Timing

3

bug(438907): SurfaceArea[ImplicitRegion[Abs[x]^n + Abs[y]^n + Abs[z]^n <= 1, {x, y, z}]]] incorrectly returns undefined for n = 3, 4, ...

SurfaceArea[ImplicitRegion[((Abs[x])^3)+((Abs[y])^3)+((Abs[z])^3)<=1,{x,y,z}]]

SupersphereAreaBaillie[3]

SupersphereAreaTrott[3]

Trott:

$Version

With[{n=3},48(∫_0^(π/4))(∫_0^(ArcSec[(Sqrt[2+((Tan[φ])^2)])]))(Sqrt[((((Cos[ϑ])^(2 n-2)) ((Sin[ϑ])^2)+((Sin[ϑ])^(2 n)) (((Cos[φ])^(2 n-2))+((Sin[φ])^(2 n-2))))/(((((Cos[ϑ])^n)+((Sin[ϑ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^(2 (n+2)/n)))])\[DifferentialD]ϑ\[DifferentialD]φ]//Timing

Baillie:

$Version

With[{k=3},Module[{a=(3^(-1/k)),b=(2^(-1/k))},Integrate[(Sqrt[1+(((x^(2k-2))+(y^(2k-2)))/(((1-(x^k)-(y^k)))^((2k-2)/k)))]),{x,0,a},{y,0,x}]]]//Timing

With[{k=3},Module[{a=(3^(-1/k)),b=(2^(-1/k))},Integrate[(Sqrt[1+(((x^(2k-2))+(y^(2k-2)))/(((1-(x^k)-(y^k)))^((2k-2)/k)))]),{x,a,b},{y,0,(((1-2(x^k)))^(1/k))}]]]//Timing

4

bug(438907): SurfaceArea[ImplicitRegion[Abs[x]^n + Abs[y]^n + Abs[z]^n <= 1, {x, y, z}]]] incorrectly returns undefined for n = 3, 4, ...

SurfaceArea[ImplicitRegion[((Abs[x])^4)+((Abs[y])^4)+((Abs[z])^4)<=1,{x,y,z}]]

SurfaceArea[ImplicitRegion[((Abs[x])^4)+((Abs[y])^4)+((Abs[z])^4)<=1,{x,y,z}],WorkingPrecision->20]

SupersphereAreaBaillie[4]

SupersphereAreaTrott[4]

Baillie:

$Version

With[{k=4},Module[{a=(3^(-1/k)),b=(2^(-1/k))},Integrate[(Sqrt[1+(((x^(2k-2))+(y^(2k-2)))/(((1-(x^k)-(y^k)))^((2k-2)/k)))]),{x,0,a},{y,0,x}]]]//Timing

With[{k=4},Module[{a=(3^(-1/k)),b=(2^(-1/k))},Integrate[(Sqrt[1+(((x^(2k-2))+(y^(2k-2)))/(((1-(x^k)-(y^k)))^((2k-2)/k)))]),{x,a,b},{y,0,(((1-2(x^k)))^(1/k))}]]]//Timing

Trott:

Michael Trott, Jan. 29, 2022

Go into spherical coordinates.
Use the general formula for the surface area of a parametrized surface.
Do the phi-integration symbolically as an indefinite integral.
Practise taking limit of incomplete elliptic integrals by hand to get complete elliptic integrals (this step is the most fun, so don't miss doing it yourself -:).
Simplify the resulting expression.
Do the resulting 1D integral with Gauss-Kronrod.

$Version

With[{n=4},48(∫_0^(π/4))(∫_0^(ArcSec[(Sqrt[2+((Tan[φ])^2)])]))(Sqrt[((((Cos[ϑ])^(2 n-2)) ((Sin[ϑ])^2)+((Sin[ϑ])^(2 n)) (((Cos[φ])^(2 n-2))+((Sin[φ])^(2 n-2))))/(((((Cos[ϑ])^n)+((Sin[ϑ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^(2 (n+2)/n)))])\[DifferentialD]ϑ\[DifferentialD]φ]//Timing

16 NIntegrate[(I Sqrt[2] (5+3 Cos[4 θ]) Sin[θ] (((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]) EllipticE[(4 (5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))/((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))])/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))-2 Sqrt[((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))] EllipticE[((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))/(4 (5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))]-2 I Sqrt[((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))] EllipticK[-((2 (2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]) Sin[θ]^4)/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ])))]+4 I (-1+Sqrt[1/(2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ])] Sqrt[2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]]) (((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]) EllipticE[(8 (2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]) Sin[θ]^4)/((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))])/(4 (5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))-EllipticK[(8 (2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]) Sin[θ]^4)/((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))])-(8 (2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]) EllipticK[(4 (5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))/((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))] Sin[θ]^4)/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))))/((3+Cos[4 θ])^(3/2) Sqrt[50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]]),{θ,0,π/2},WorkingPrecision->200,PrecisionGoal->100]

Integrate[(I Sqrt[2] (5+3 Cos[4 θ]) Sin[θ] (((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]) EllipticE[(4 (5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))/((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))])/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))-2 Sqrt[((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))] EllipticE[((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))/(4 (5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))]-2 I Sqrt[((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))] EllipticK[-((2 (2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]) Sin[θ]^4)/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ])))]+4 I (-1+Sqrt[1/(2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ])] Sqrt[2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]]) (((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]) EllipticE[(8 (2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]) Sin[θ]^4)/((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))])/(4 (5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))-EllipticK[(8 (2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]) Sin[θ]^4)/((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))])-(8 (2+21 Cos[2 θ]+6 Cos[4 θ]+3 Cos[6 θ]) EllipticK[(4 (5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))/((3+Cos[4 θ]) (50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]))] Sin[θ]^4)/((5+3 Cos[4 θ]) (9+4 Cos[2 θ]+3 Cos[4 θ]))))/((3+Cos[4 θ])^(3/2) Sqrt[50+45 Cos[2 θ]+30 Cos[4 θ]+3 Cos[6 θ]]),θ]//Timing

((((Cos[ϑ])^(2 n-2)) ((Sin[ϑ])^2)+((Sin[ϑ])^(2 n)) (((Cos[φ])^(2 n-2))+((Sin[φ])^(2 n-2))))/(((((Cos[ϑ])^n)+((Sin[ϑ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^((2 (2+n))/n)))/.n->4

48 NIntegrate[(Sqrt[((((Cos[ϑ])^6) ((Sin[ϑ])^2)+((Sin[ϑ])^8) (((Cos[φ])^6)+((Sin[φ])^6)))/(((((Cos[ϑ])^4)+((Sin[ϑ])^4) (((Cos[φ])^4)+((Sin[φ])^4))))^3))]),{φ,0,(π/4)},{ϑ,0,ArcCos[(1/(Sqrt[2+((Tan[φ])^2)]))]}]

FullSimplify[((((Cos[ϑ])^6) ((Sin[ϑ])^2)+((Sin[ϑ])^8) (((Cos[φ])^6)+((Sin[φ])^6)))/(((((Cos[ϑ])^4)+((Sin[ϑ])^4) (((Cos[φ])^4)+((Sin[φ])^4))))^3))]

48 NIntegrate[(Sqrt[((64 ((Csc[ϑ])^4) (((Cos[φ])^6)+((Cot[ϑ])^6)+((Sin[φ])^6)))/(((3+Cos[4 φ]+4 ((Cot[ϑ])^4)))^3))]),{φ,0,(π/4)},{ϑ,0,ArcCos[(1/(Sqrt[2+((Tan[φ])^2)]))]}]

48 8 NIntegrate[(Sqrt[((((Csc[ϑ])^4) (((Cos[φ])^6)+((Cot[ϑ])^6)+((Sin[φ])^6)))/(((3+Cos[4 φ]+4 ((Cot[ϑ])^4)))^3))]),{φ,0,(π/4)},{ϑ,0,ArcCos[(1/(Sqrt[2+((Tan[φ])^2)]))]}]

384 NIntegrate[((Csc[ϑ])^2) (Sqrt[((((Cos[φ])^6)+((Cot[ϑ])^6)+((Sin[φ])^6))/(((3+Cos[4 φ]+4 ((Cot[ϑ])^4)))^3))]),{φ,0,(π/4)},{ϑ,0,ArcSec[(Sqrt[2+((Tan[φ])^2)])]}]

Assuming[0<φ<π/4,Integrate[((Csc[ϑ])^2) (Sqrt[((((Cos[φ])^6)+((Cot[ϑ])^6)+((Sin[φ])^6))/(((3+Cos[4 φ]+4 ((Cot[ϑ])^4)))^3))]),{ϑ,0,ArcSec[(Sqrt[2+((Tan[φ])^2)])]}]]//Timing

FullSimplify[((Csc[ϑ])^2) (Sqrt[((((Cos[φ])^6)+((Cot[ϑ])^6)+((Sin[φ])^6))/(((3+Cos[4 φ]+4 ((Cot[ϑ])^4)))^3))])(2/(1+(t^2)))/.{Cos[ϑ]->((1-(t^2))/(1+(t^2))),Csc[ϑ]->((1+(t^2))/(2t)),Cot[ϑ]->(((1-(t^2))/(1+(t^2)))/((2t)/(1+(t^2))))}]

Tan[{0,ArcSec[(Sqrt[2+((Tan[φ])^2)])]}/2]//FullSimplify

384NIntegrate[(((1+(t^2)) (Sqrt[((((((-1+(t^2)))^6)/(64 (t^6)))+((Cos[φ])^6)+((Sin[φ])^6))/(((3+((((-1+(t^2)))^4)/(4 (t^4)))+Cos[4 φ]))^3))]))/(2 (t^2))),{φ,0,(π/4)},{t,0,Tan[(1/2) ArcSec[(Sqrt[2+((Tan[φ])^2)])]]}]

Together[((((((-1+(t^2)))^6)/(64 (t^6)))+((Cos[φ])^6)+((Sin[φ])^6))/(((3+((((-1+(t^2)))^4)/(4 (t^4)))+Cos[4 φ]))^3))]

384NIntegrate[(((1+(t^2)) )/(2(t^2)))(Sqrt[(((t^6) (1-6 (t^2)+15 (t^4)-20 (t^6)+15 (t^8)-6 (t^10)+(t^12)+64 (t^6) ((Cos[φ])^6)+64 (t^6) ((Sin[φ])^6)))/(((1-4 (t^2)+18 (t^4)-4 (t^6)+(t^8)+4 (t^4) Cos[4 φ]))^3))]),{φ,0,(π/4)},{t,0,Tan[(1/2) ArcSec[(Sqrt[2+((Tan[φ])^2)])]]}]

ResourceFunction["IntegrateAlgebraic"][(1/2)(1+(t^2))t (Sqrt[(( 1-6 (t^2)+15 (t^4)-20 (t^6)+15 (t^8)-6 (t^10)+(t^12)+64 (t^6) ((Cos[φ])^6)+64 (t^6) ((Sin[φ])^6))/(((1-4 (t^2)+18 (t^4)-4 (t^6)+(t^8)+4 (t^4) Cos[4 φ]))^3))]),t]

384NIntegrate[(1/2)(1+(t^2))t (Sqrt[(( 1-6 (t^2)+15 (t^4)-20 (t^6)+15 (t^8)-6 (t^10)+(t^12)+64 (t^6) ((Cos[φ])^6)+64 (t^6) ((Sin[φ])^6))/(((1-4 (t^2)+18 (t^4)-4 (t^6)+(t^8)+4 (t^4) Cos[4 φ]))^3))]),{φ,0,(π/4)},{t,0,Tan[(1/2) ArcSec[(Sqrt[2+((Tan[φ])^2)])]]}]

FullSimplify[Tan[(1/2) ArcSec[(Sqrt[2+((Tan[φ])^2)])]],0<φ<π/4]

384Integrate[(1/2)(1+(t^2))t (Sqrt[(( 1-6 (t^2)+15 (t^4)-20 (t^6)+15 (t^8)-6 (t^10)+(t^12)+64 (t^6) ((Cos[φ])^6)+64 (t^6) ((Sin[φ])^6))/(((1-4 (t^2)+18 (t^4)-4 (t^6)+(t^8)+4 (t^4) Cos[4 φ]))^3))]),t]//Timing

Integrate[(1/2)(1+(t^2))t (Sqrt[(( 1-6 (t^2)+15 (t^4)-20 (t^6)+15 (t^8)-6 (t^10)+(t^12)+64 (t^6) ((Cos[φ])^6)+64 (t^6) ((Sin[φ])^6))/(((1-4 (t^2)+18 (t^4)-4 (t^6)+(t^8)+4 (t^4) Cos[4 φ]))^3))]),t]//Timing

384Integrate[(1/2)(1+(t^2))t (Sqrt[(( 1-6 (t^2)+15 (t^4)-20 (t^6)+15 (t^8)-6 (t^10)+(t^12)+64 (t^6) ((Cos[φ])^6)+64 (t^6) ((Sin[φ])^6))/(((1-4 (t^2)+18 (t^4)-4 (t^6)+(t^8)+4 (t^4) Cos[4 φ]))^3))]),{t,0,Tan[(1/2) ArcSec[(Sqrt[2+((Tan[φ])^2)])]]}]//Timing

n

Assuming[n>0,SurfaceArea[ImplicitRegion[((Abs[x])^n)+((Abs[y])^n)+((Abs[z])^n)<=1,{x,y,z}]]]//Timing

∞ == cube

Volume

1 == octahedron

Volume[ImplicitRegion[Abs[x]+Abs[y]+Abs[z]<=1,{x,y,z}]]

2 == sphere

Volume[ImplicitRegion[((Abs[x])^2)+((Abs[y])^2)+((Abs[z])^2)<=1,{x,y,z}]]

4

Volume[ImplicitRegion[((Abs[x])^4)+((Abs[y])^4)+((Abs[z])^4)<=1,{x,y,z}]]

FunctionExpand[%]

N[%]

FullSimplify[Gamma[(5/4)]-(1/4)Gamma[(1/4)]]

FullSimplify[Subtract@@{Gamma[-(1/4)],-(π/((1/4)Sin[(π/4)]Gamma[(1/4)]))}]

FullSimplify[-((8 ((Gamma[(1/4)])^2) (1/4)Gamma[(1/4)])/(-3(π/((1/4)Sin[(π/4)]Gamma[(1/4)]))))]

N[%]

6 == Hauser's "cube"

FullSimplify[par={((a Cos[v] Sin[u])/(((((Cos[u])^6)+((Sin[u])^6) (((Cos[v])^6)+((Sin[v])^6))))^(1/6))),((a Sin[u] Sin[v])/(((((Cos[u])^6)+((Sin[u])^6) (((Cos[v])^6)+((Sin[v])^6))))^(1/6))),((a Cos[u])/(((((Cos[u])^6)+((Sin[u])^6) (((Cos[v])^6)+((Sin[v])^6))))^(1/6)))}]

ChristoffelSymbolOfTheSecondKind

FullSimplify[ChristoffelSymbols[par,{u,v}],a>0&&0<u<2π&&0<v<π]//Timing

First Fundamental Form

FullSimplify[FirstFundamentalForm[par,{u,v}],a>0&&0<u<2π&&0<v<π]//Timing

GaussianCurvature

FullSimplify[GaussianCurvature[par,{u,v}],a>0&&0<u<2π&&0<v<π]//Timing

MeanCurvature

FullSimplify[MeanCurvature[par,{u,v}],a>0&&0<u<2π&&0<v<π]//Timing

Metric

FullSimplify[MetricTensor[par,{u,v}],a>0&&0<u<2π&&0<v<π]

PrincipalCurvatures

PrincipalCurvatures[par,{u,v}]//Timing

Simplify[PrincipalCurvatures[par,{u,v}],a>0&&0<u<2π&&0<v<π]//Timing

FullSimplify[PrincipalCurvatures[par,{u,v}],a>0&&0<u<2π&&0<v<π]//Timing

Second Fundamental Form

FullSimplify[SecondFundamentalForm[par,{u,v}],a>0&&0<u<2π&&0<v<π]//Timing

SurfaceArea

{e,f,g}={(((a^2) (((Cos[u])^10)+(1/64) (((5+3 Cos[4 v]))^2) ((Sin[u])^10)))/(((((Cos[u])^6)+((Sin[u])^6) (((Cos[v])^6)+((Sin[v])^6))))^(7/3))),(((a^2) ((Sin[u])^7) ((70 Cos[u]+55 Cos[3 u]+3 Cos[5 u]) Sin[4 v]-24 Cos[u] ((Sin[u])^4) Sin[8 v]))/(512 (((((Cos[u])^6)+((Sin[u])^6) (((Cos[v])^6)+((Sin[v])^6))))^(7/3)))),((a^2) ((Sin[u])^2) (((Cos[u])^12)+(1/4) ((Cos[u])^6) (5+3 Cos[4 v]) ((Sin[u])^6)+(1/128) (63+60 Cos[4 v]+5 Cos[8 v]) ((Sin[u])^12)+(1/16) ((Cos[u])^2) ((Sin[u])^10) ((Sin[4 v])^2)))/(((((Cos[u])^6)+((Sin[u])^6) (((Cos[v])^6)+((Sin[v])^6))))^(7/3))};

NIntegrate[(Sqrt[e g-f^2])/.a->1,{u,0,2π},{v,0,π}]

FullSimplify[(Sqrt[e g-f^2]),a>0&&0<u<2π&&0<v<π]//Timing

(1/512)NIntegrate[√((((Sin[u])^2) (262144 (((Cos[u])^10)+(1/64) (((5+3 Cos[4 v]))^2) ((Sin[u])^10)) (((Cos[u])^12)+(1/4) ((Cos[u])^6) (5+3 Cos[4 v]) ((Sin[u])^6)+(1/128) (63+60 Cos[4 v]+5 Cos[8 v]) ((Sin[u])^12)+(1/16) ((Cos[u])^2) ((Sin[u])^10) ((Sin[4 v])^2))-((Sin[u])^12) ((((70 Cos[u]+55 Cos[3 u]+3 Cos[5 u]) Sin[4 v]-24 Cos[u] ((Sin[u])^4) Sin[8 v]))^2)))/(((((Cos[u])^6)+((Sin[u])^6) (((Cos[v])^6)+((Sin[v])^6))))^(14/3))),{u,0,2π},{v,0,π}]

Volume

Assuming[a>0,With[{n=6},Integrate[(r^2) (((((Cos[θ])^n)+((Sin[θ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^(-2/n))Sin[φ],{r,0,a},{θ,0,2π},{φ,0,π}]]]//Timing

Assuming[a>0,With[{n=6},Integrate[(r^2) (((((Cos[θ])^n)+((Sin[θ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^(-2/n))Sin[φ],{θ,0,2π},{φ,0,π},{r,0,a}]]]//Timing

Volume[ImplicitRegion[((Abs[x])^6)+((Abs[y])^6)+((Abs[z])^6)<=1,{x,y,z}]]//Timing

-((32 (Sqrt[π]) ((Gamma[(7/6)])^2))/(Gamma[-(1/6)]))//N

((8((Gamma[1+(1/n)])^3))/(Gamma[1+(3/n)]))/.n->6

N[%]

Γ(1+z)==z Γ(z)

((16 ((((1/6)Gamma[(1/6)]))^3))/(Sqrt[π]))//N

Simplify[((16 ((((1/6)Gamma[(1/6)]))^3))/(Sqrt[π]))]

n

V4.2

Integrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))},Assumptions->n>0&&n∈Integers]//Timing

With[{n=Pi//N},NIntegrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))}]]

FullSimplify[(((1-(x^n)))^(2/n)) Hypergeometric2F1[-(1/n),(1/n),1+(1/n),-(((((((1-(x^n)))^(1/n))))^n)/(-1+(x^n)))],n>0&&n∈Integers]

Integrate[(((4^(-1/n)) (Sqrt[π]) (((1-(x^n)))^(2/n)) Gamma[1+(1/n)])/(Gamma[(1/2)+(1/n)])),{x,0,1},Assumptions->n>0&&n∈Integers]//Timing

%/.n->Pi//N

With[{n=Pi//N},
NIntegrate[(((4^(-1/n)) (Sqrt[π]) (((1-(x^n)))^(2/n)) Gamma[1+(1/n)])/(Gamma[(1/2)+(1/n)])),{x,0,1}]]

FullSimplify[(((4^(-1/n)) (Sqrt[π]) Gamma[1+(1/n)] Gamma[1+(2/n)] Gamma[(1/n)])/(3 Gamma[(1/2)+(1/n)] Gamma[(3/n)])),n∈Integers&&n>0]

expr=FullSimplify[Subtract@@{((((Gamma[1+(1/n)])^2) Gamma[(1/n)])/(3 Gamma[(3/n)])),(((Gamma[1+(1/n)])^3)/(Gamma[1+(3/n)]))},n∈Integers&&n>0]

Table[expr/.n->Random[Complex,10(I+1){-1,1},20],{100}]//Chop//Union

V5.0

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

Integrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))},Assumptions->n∈Integers&&n>0]//Timing

With[{n=Pi},NIntegrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))}]]

0.9999999999999999999999999999`33^Pi

FullSimplify[(((1-(x^n)))^(1/n)) (((1-(x^n)-((((((1-(x^n)))^(1/n))))^n)))^(1/n)) (((((-1+(x^n)+((((((1-(x^n)))^(1/n))))^n))/(-1+(x^n)))))^(-1/n)) Hypergeometric2F1[(1/n),-(1/n),1+(1/n),(((((((1-(x^n)))^(1/n))))^n)/(1-(x^n)))],n∈Integers&&n>0]

<<Calculus`Integration`

With[{a=1,b=1,c=1},Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->n>0]
]//Timing

With[{a=1,b=1,c=1},Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]]//Timing

Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0&&c>0&&n>0]//Timing

v5.0

<<Calculus`Integration`

Table[Integrate[Boole[Abs[x]^n+Abs[y]^n+Abs[z]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}],{n,5}]//Timing

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

FullSimplify[%]

N[%]//Chop

Table[(((Gamma[1+(1/n)])^3)/(Gamma[((3+n)/n)])),{n,5}]//FullSimplify

N[%]

<<Utilities`Simplify`

GammaSimplify[{(1/6),(π/6),((Gamma[(4/3)])^3),(((Gamma[(5/4)])^3)/(Gamma[(7/4)])),(((Gamma[(6/5)])^3)/(Gamma[(8/5)]))}]//FullSimplify

v5.1

Table[Integrate[Boole[Abs[x]^n+Abs[y]^n+Abs[z]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}],{n,5}]//Timing

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

FullSimplify[%]

N[%]//Chop

Table[(((Gamma[1+(1/n)])^3)/(Gamma[((3+n)/n)])),{n,5}]//FullSimplify

N[%]

<<Utilities`Simplify`

GammaSimplify[{(1/6),(π/6),((Gamma[(4/3)])^3),(((Gamma[(5/4)])^3)/(Gamma[(7/4)])),(((Gamma[(6/5)])^3)/(Gamma[(8/5)]))}]//FullSimplify

V5.1

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

Integrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))},Assumptions->n>0&&n∈Integers]//Timing

With[{n=Pi},NIntegrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))}]]//Chop

0.9999999999999999999999999999`33^Pi

With[{n=Pi//N},NIntegrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))}]]//Chop

(((Gamma[1+(1/n)])^3)/(Gamma[((3+n)/n)]))/.n->Pi//N

With[{a=1,b=1,c=1},Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->n>0&&n∈Integers]
]//Timing

With[{a=1,b=1,c=1},Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]]//Timing

Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0&&c>0&&n>0]//Timing

V5.2

$Version

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

Integrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))},Assumptions->n>0&&n∈Integers]//Timing

Table[((((Gamma[1+(1/n)])^2) Gamma[(1/n)])/(3 Gamma[(3/n)])),{n,5}]//N

With[{a=1,b=1,c=1},Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]]//Timing

Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0&&c>0&&n>0]//Timing

V6

$Version

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

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

Table[(((Gamma[1+(1/n)])^3)/(Gamma[1+(3/n)])),{n,5}]//N

Integrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))},Assumptions->n>0&&n∈Integers]//Timing

FullSimplify[(((4^(-1/n)) (π^(5/2)) Csc[(π/n)] Csc[((2 π)/n)] Gamma[1+(1/n)])/(Gamma[(1/2)+(1/n)] Gamma[-(2/n)] Gamma[-(1/n)] Gamma[((3+n)/n)])),n∈Integers&&n>2]

Table[(((4^(-1/n)) (π^(5/2)) Csc[(π/n)] Csc[((2 π)/n)] Gamma[1+(1/n)])/(Gamma[(1/2)+(1/n)] Gamma[-(2/n)] Gamma[-(1/n)] Gamma[((3+n)/n)])),{n,5}]//N

With[{a=1,b=1,c=1},Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]]//Timing

Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0&&c>0&&n>0]//Timing

Table[Integrate[Boole[x^n+y^n+z^n<a^n],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0],{n,2,6}]//Timing

Table[Integrate[Boole[Abs[x]^n+Abs[y]^n+Abs[z]^n<a^n],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0],{n,2,6}]//Timing

V7

$Version

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

Integrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))},Assumptions->n>0&&n∈Integers]//Timing

Integrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))},Assumptions->n>0&&n∈Integers,GenerateConditions->True]//Timing

With[{a=1,b=1,c=1},Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]]//Timing

Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0&&c>0&&n>0]//Timing

Table[Integrate[Boole[x^n+y^n+z^n<a^n],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0],{n,2,6}]//Timing

V8

$Version

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

Integrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))},Assumptions->n>0&&n∈Integers]//Timing

Integrate[(((1-(x^n)-(y^n)))^(1/n)),{x,0,1},{y,0,(((1-(x^n)))^(1/n))},Assumptions->n>0&&n∈Integers,GenerateConditions->True]//Timing

With[{a=1,b=1,c=1},Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]]//Timing

Integrate[Boole[Abs[x/a]^n+Abs[y/b]^n+Abs[z/c]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0&&c>0&&n>0]//Timing

Table[Integrate[Boole[x^n+y^n+z^n<a^n],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0],{n,2,6}]//Timing

8Table[Integrate[Boole[Abs[x]^n+Abs[y]^n+Abs[z]^n<a^n],{x,0,∞},{y,0,∞},{z,0,∞},Assumptions->a>0],{n,2,6}]//Timing

V13.1

$Version

Assuming[n>0,Volume[ImplicitRegion[((Abs[x])^n)+((Abs[y])^n)+((Abs[z])^n)<=1,{x,y,z}]]]//Timing

Do[
Print[{n,TimeConstrained[Volume[ImplicitRegion[Abs[x]^n+Abs[y]^n+Abs[z]^n<1,{x,y,z}]],60]//Timing}],
{n,6}]

Assuming[n>0,Integrate[Boole[Abs[x]^n+Abs[y]^n+Abs[z]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]]//Timing

bug(419859): Assuming[n > 0, Integrate[Boole[Abs[x]^n + Abs[y]^n + Abs[z]^n < 1], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}, {z, -Infinity, Infinity}]] flips inequality in unevaluated result

TimeConstrained[Integrate[Boole[Abs[x]^n+Abs[y]^n+Abs[z]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}],60]//Timing

Do[
Print[{n,TimeConstrained[Integrate[Boole[Abs[x]^n+Abs[y]^n+Abs[z]^n<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}],60]//Timing}],
{n,6}]

V14

$Version

Assuming[n>0,Volume[ImplicitRegion[((Abs[x])^n)+((Abs[y])^n)+((Abs[z])^n)<=1,{x,y,z}]]]//Timing

Do[
Print[{n,TimeConstrained[Volume[ImplicitRegion[Abs[x]^n+Abs[y]^n+Abs[z]^n<1,{x,y,z}]],60]//Timing}],
{n,6}]

∞ == cube

Limit[((8((Gamma[1+(1/n)])^3))/(Gamma[((3+n)/n)])),n->∞]

Series of volumes

Table[((8((Gamma[(1+(1/n))])^3))/(Gamma[1+(3/n)])),{n,6}]//FullSimplify

Plot[((8((Gamma[(1+(1/n))])^3))/(Gamma[1+(3/n)])),{n,.01,10},Epilog->Point[Table[{n,((8((Gamma[(1+(1/n))])^3))/(Gamma[1+(3/n)]))},{n,10}]]]

Prudnikov Integral

V6

Integrate[(x^(α-1))(y^(β-1))(z^(γ-1))Boole[(x/a)^p+(y/b)^q+(z/c)^r≤1],{x,0,∞},{y,0,∞},{z,0,∞}]

With[{a=1,b=1,c=1,p=n,q=n,r=n},Integrate[(x^(α-1))(y^(β-1))(z^(γ-1))Boole[(x/a)^p+(y/b)^q+(z/c)^r≤1],{x,0,∞},{y,0,∞},{z,0,∞}]
]

With[{β=γ=α=1,p=q=r=n,a=b=c=1},
{
{α,β,γ,a,b,c,p,q,r},
{
Integrate[(x^(α-1))(y^(β-1))(z^(γ-1))Boole[(x/a)^p+(y/b)^q+(z/c)^r≤1],{x,0,∞},{y,0,∞},{z,0,∞}],
(((a^α)(b^β)(c^γ))/(p q r))((Gamma[(α/p)]Gamma[(β/q)]Gamma[(γ/r)])/(Gamma[(α/p)+(β/q)+(γ/r)+1]))
}
}]

Table[Last[%%],{n,5}]

N[%]//Chop

Inertia tensor

Table[(M/(((8((Gamma[(1+(1/n))])^3))/(Gamma[1+(3/n)]))(a^3)))Integrate[InertiaTensor[{x,y,z}]Boole[x^n+y^n+z^n<a^n],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0],{n,2,10,2}]//FullSimplify

MatrixForm/@GammaSimplify[%]//TraditionalForm

FullSimplify[%%]

Spherical coordinate parametrization

With[{a=1,n=5},ParametricPlot3D[a/(Abs[Cos[θ]]^n+Abs[Sin[θ]]^n (Abs[Cos[φ]]^n+Abs[Sin[φ]]^n))^(1/n) {Cos[φ] Sin[θ],Sin[φ] Sin[θ],Cos[θ]},{φ,0,2 Pi},{θ,0,Pi},Boxed->False,Axes->False]]

{x,y,z}=a/(Cos[θ]^6+Sin[θ]^6 (Cos[φ]^6+Sin[φ]^6))^(1/6) {Cos[φ] Sin[θ],Sin[φ] Sin[θ],Cos[θ]}

{((a Cos[φ] Sin[θ])/(((((Cos[θ])^6)+((Sin[θ])^6) (((Cos[φ])^6)+((Sin[φ])^6))))^(1/6))),((a Sin[θ] Sin[φ])/(((((Cos[θ])^6)+((Sin[θ])^6) (((Cos[φ])^6)+((Sin[φ])^6))))^(1/6))),((a Cos[θ])/(((((Cos[θ])^6)+((Sin[θ])^6) (((Cos[φ])^6)+((Sin[φ])^6))))^(1/6)))}/.{φ->v,θ->u}

FullSimplify[x^6+y^6+z^6]

{x,y,z}=a/(Cos[θ]^n+Sin[θ]^n (Cos[φ]^n+Sin[φ]^n))^(1/n) {Cos[φ] Sin[θ],Sin[φ] Sin[θ],Cos[θ]}

FullSimplify[x^n+y^n+z^n,Assumptions->n>0&&n∈Integers]

FullSimplify[x^n+y^n+z^n,Assumptions->n>0&&n∈Integers&&Mod[n,2]==0]

Table[FullSimplify[x^n+y^n+z^n],{n,8}]

With[{r=RandomReal[{0,5}]},x^n+y^n+z^n-a^n/.{x->RandomReal[{-1,1}r],y->RandomReal[{-1,1}r],z->RandomReal[{-1,1}r],n->4}]

Volume

r2=Simplify[#.#&[a/(Cos[θ]^n+Sin[θ]^n (Cos[φ]^n+Sin[φ]^n))^(1/n) {Cos[φ] Sin[θ],Sin[φ] Sin[θ],Cos[θ]}]]

With[{n=2},Integrate[(r^2) (((((Cos[θ])^n)+((Sin[θ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^(-2/n))Sin[φ],{r,0,a},{θ,0,2π},{φ,0,π}]]

Assuming[a>0,With[{n=4},Integrate[(r^2) (((((Cos[θ])^n)+((Sin[θ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^(-2/n))Sin[φ],{r,0,a},{θ,0,2π},{φ,0,π}]]]//Timing

Assuming[a>0,With[{n=6},Integrate[(r^2) (((((Cos[θ])^n)+((Sin[θ])^n) (((Cos[φ])^n)+((Sin[φ])^n))))^(-2/n))Sin[φ],{r,0,a},{θ,0,2π},{φ,0,π}]]]//Timing