Watson's Triple Integrals

Author

Eric W. Weisstein
May 20, 2011

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

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

©2011 Wolfram Research, Inc. except for portions noted otherwise

(Subscript[I, 1])==(∫_0^π)(∫_0^π)(∫_0^π)(1/(1-cos(u) cos(v) cos(w)))dwdvdu==(((Gamma[(1/4)])^4)/(4 (π^3)))

Numerical

(1/(π^3))NIntegrate[(1/(1-Cos[u]Cos[v]Cos[w])),{u,0,π},{v,0,π},{w,0,π}]//Timing

(1/(π^3))NIntegrate[(1/(1-Cos[u]Cos[v]Cos[w])),{u,0,π},{v,0,π},{w,0,π},Method->QuasiMonteCarlo,PrecisionGoal->3,WorkingPrecision->10]//Timing

N[{((4((EllipticK[((((1/2)(Sqrt[2])))^2)])^2))/(π^2)),(((Gamma[(1/4)])^4)/(4(π^3)))},20]

Analytic

<<Utilities`Simplify`

Watson Transformation

i1=(4/(π^3))Integrate[((Sin[θ])/(1+(1/4)(r^4)((Sin[θ])^4)((Cos[θ])^2)((Sin[ψ])^2))),{ψ,0,π/2},{θ,0,π/2},{r,0,∞}]

i1//EllipticSimplify//GammaSimplify//TraditionalForm

{((4(Sqrt[2]))/(π^3)),Integrate[(1/(1+(t^4))),{t,0,∞}],Integrate[(1/(Sqrt[Cos[θ]])),{θ,0,(π/2)}],Integrate[(1/(Sqrt[Sin[ψ]])),{ψ,0,(π/2)}]}

%//EllipticSimplify//GammaSimplify

Times@@%

EllipticSimplify[%]

Direct Integration

Integrate[(1/(1-Cos[u]Cos[v]Cos[w])),{u,0,π},{v,0,π},{w,0,π}]//Timing

(1/(π^2))NIntegrate[(1/(Sqrt[1-Cos[v]^2Cos[w]^2])),{v,0,π},{w,0,π},Method->QuasiMonteCarlo,PrecisionGoal->3,MaxPoints->10^6]

(1/(π^2))NIntegrate[(1/(Sqrt[1-Cos[v]^2Cos[w]^2])),{v,0,π},{w,0,π},WorkingPrecision->15]

Integrate[(1/(1-Cos[u] x)),{u,0,π},Assumptions->-1<x<1]

%/.x->Cos[v]Cos[w]

FullSimplify[Integrate[((Cos[v])^(2n)),{v,0,π}],n∈Integers]

Table[(((Sqrt[π]) Gamma[(1/2)+n])/(Gamma[1+n]))-((π(2n-1)!!)/((2n)!!)),{n,5}]

Sum[(((2n-1)!!)/((2n)!!)),{n,0,∞}]

Sum[FunctionExpand[((((((2n-1)!!)/((2n)!!))))^3)],{n,0,∞}]//GammaSimplify

v5.0.1

Integrate[(1/(1-Cos[u]Cos[v]Cos[w])),{u,0,π},Assumptions->And@@(0<#<Pi&/@{v,w})]

v6

Integrate[(1/(1-Cos[u]Cos[v]Cos[w])),{u,0,π},Assumptions->And@@(0<#<Pi&/@{v,w})]

DI-2

FullSimplify[Integrate[1/(1-Cos[u]*Cos[v]*Cos[w]),{u,0,Pi},Assumptions->And@@(0<#1<Pi&)/@{v,w}],And@@(0<#1<Pi&)/@{v,w}]

NIntegrate[(2*Abs[Cos[v]]*Abs[Cos[w]]*(Log[-1-Sec[v]*Sec[w]]-Log[1+Sec[v]*Sec[w]])*Sec[v]*Sec[w])/Sqrt[-3+Cos[2*v]+2*Cos[v]^2*Cos[2*w]],{v,0,π},{w,0,π},Method->QuasiMonteCarlo]/π^3

Integrate[(2*Abs[Cos[v]]*Abs[Cos[w]]*(Log[-1-Sec[v]*Sec[w]]-Log[1+Sec[v]*Sec[w]])*Sec[v]*Sec[w])/Sqrt[-3+Cos[2*v]+2*Cos[v]^2*Cos[2*w]],{v,0,π},Assumptions->0<w<π]

V8.0

$Version

(i=Integrate[(1/(1-Cos[u]Cos[v]Cos[w])),{u,0,π},{v,0,π},{w,0,π},Assumptions->(0<#<π&/@{u,v,w})])//Timing

(i=Integrate[1/(1-Cos[u] Cos[v] Cos[w]),{u,0,Pi},Assumptions->0<v<Pi&&0<w<Pi])//Timing

(1/(π^3))NIntegrate[((2 I π)/(Sqrt[-3+2 Cos[2 v] ((Cos[w])^2)+Cos[2 w]])),{v,0,π},{w,0,π}]

(i=Integrate[((2 I π)/(Sqrt[-3+2 Cos[2 v] ((Cos[w])^2)+Cos[2 w]])),{v,0,π},Assumptions->0<w<π])//Timing

(1/(π^3))NIntegrate[π (EllipticK[((Cos[w])^2)]+Csc[w] EllipticK[-((Cot[w])^2)]),{w,0,π}]

(((Gamma[(1/4)])^4)/(4(π^3)))//N

FullSimplify[π (EllipticK[((Cos[w])^2)]+Csc[w] EllipticK[-((Cot[w])^2)]),0<w<π]

Integrate[π (EllipticK[((Cos[w])^2)]+Csc[w] EllipticK[-((Cot[w])^2)]),{w,0,π}]//Timing

Integrate[π (EllipticK[((Cos[w])^2)]+Csc[w] EllipticK[-((Cot[w])^2)]),w]//Timing

(i=Integrate[(1/(1-Cos[u]Cos[v]Cos[w])),{u,0,π},{v,0,π},{w,0,π}])//Timing

V8.1

$Version

(i=Integrate[1/(1-Cos[u] Cos[v] Cos[w]),{u,0,Pi},Assumptions->0<v<Pi&&0<w<Pi])//Timing

(1/(π^3))NIntegrate[((2 I π)/(Sqrt[-3+2 Cos[2 v] ((Cos[w])^2)+Cos[2 w]])),{v,0,π},{w,0,π}]

Integrate[((2 I π)/(Sqrt[-3+2 Cos[2 v] ((Cos[w])^2)+Cos[2 w]])),{v,0,π},Assumptions->0<w<π]

(i=Integrate[(1/(1-Cos[u]Cos[v]Cos[w])),{u,0,π},{v,0,π},{w,0,π}])//Timing

(i=Integrate[(1/(1-Cos[u]Cos[v]Cos[w])),{u,0,π},{v,0,π},{w,0,π},Assumptions->(0<#<π&/@{u,v,w})])//Timing

Sums

(∑_(n=0)^∞)((((((2 n-1)!!)/((2 n)!!))))^3)==(((Gamma[(1/4)])^4)/(4 (π^3)))==(π/((Gamma[(3/4)])^4))

Sum[((((((2n-1)!!)/((2n)!!))))^3),{n,0,∞}]

FullSimplify[%]

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

4((EllipticK[(1/2)])^2)/(π^2)//FunctionExpand

(Subscript[I, 2])==(∫_0^π)(∫_0^π)(∫_0^π)(1/(-cos(u) cos(v)-cos(u) cos(w)-cos(v) cos(w)+3))dwdvdu

Numerical

(1/(π^3))NIntegrate[(1/(3-Cos[v]Cos[w]-Cos[w]Cos[u]-Cos[u]Cos[v])),{u,0,π},{v,0,π},{w,0,π}]//Timing

(1/(π^3))NIntegrate[(1/(3-Cos[v]Cos[w]-Cos[w]Cos[u]-Cos[u]Cos[v])),{u,0,π},{v,0,π},{w,0,π},Method->QuasiMonteCarlo,PrecisionGoal->2,WorkingPrecision->10]//Timing

N[{(((Sqrt[3])((EllipticK[((((1/4)((Sqrt[6])-(Sqrt[2]))))^2)])^2))/(π^2)),((3((Gamma[(1/3)])^6))/((2^(14/3))(π^4)))},20]

Analytic

v5.1

(1/(π^3))Integrate[(1/(3-Cos[v]Cos[w]-Cos[w]Cos[u]-Cos[u]Cos[v])),{u,0,π},{v,0,π},{w,0,π}]//Timing

v5.2

(1/(π^3))Integrate[(1/(3-Cos[v]Cos[w]-Cos[w]Cos[u]-Cos[u]Cos[v])),{u,0,π},{v,0,π},{w,0,π}]//Timing

Watson

v5

(1/(π^2))Integrate[(1/(Sqrt[(1+(t^2))(1+(1/4)(t^2)((Sin[ψ])^2))])),{ψ,0,(π/2)},{t,0,∞}]

(1/(π^2))Integrate[(1/(Sqrt[(1+(t^2))(1+(1/4)(t^2)((Sin[ψ])^2))])),{t,0,∞},{ψ,0,(π/2)}]

(1/(π^2))Integrate[(1/(Sqrt[(1+(t^2))(1+(1/4)(t^2)((Sin[ψ])^2))])),{ψ,0,(π/2)},Assumptions->t>0]

Integrate[((EllipticK[-((t^2)/4)])/((π^2) (Sqrt[1+(t^2)]))),{t,0,∞}]

2Integrate[(((1/(Sqrt[(u^2)+1]))EllipticK[((u^2)/((u^2)+1))])/((π^2) (Sqrt[1+4(u^2)]))),{u,0,∞}]

2NIntegrate[(((1/(Sqrt[(u^2)+1]))EllipticK[((u^2)/((u^2)+1))])/((π^2) (Sqrt[1+4(u^2)]))),{u,0,10^9},MaxRecursion->20,WorkingPrecision->20]

(1/(π^2))Integrate[EllipticK[1-(1/4)((Sin[ψ])^2)],{ψ,0,(π/2)}]

(1/(π^2))NIntegrate[EllipticK[1-(1/4)((Sin[ψ])^2)],{ψ,0,(π/2)}]

DI-2

In[1]:= $Pre=InputForm

Out[1]= InputForm

In[2]:= Integrate[1/Sqrt[(1 + t^2)*(1 + (1/4)*t^2*Sin[ψ]^2)], {ψ, 0, Pi/2}, {t, 0, Infinity}]/Pi^2

Out[2]//InputForm= 
Integrate[MeijerG[{{1, 1}, {}}, {{1/2, 1/2}, {}}, 4*Csc[ψ]^2]/(2*Pi), {ψ, 0, Pi/2}]/Pi^2

In[3]:= Integrate[1/Sqrt[(1 + t^2)*(1 + (1/4)*t^2*Sin[ψ]^2)], {t,0,Infinity},{ψ,0,Pi/2}]/Pi^2

Out[3]//InputForm= 
MeijerG[{{1/2, 1/2}, {1/2}}, {{0, 0, 0}, {}}, 4]/(2*Pi^(5/2))

TraditionalForm[MeijerG[{{1/2,1/2},{1/2}},{{0,0,0},{}},4]/(2*Pi^(5/2))]

N[MeijerG[{{1/2,1/2},{1/2}},{{0,0,0},{}},4]/(2*Pi^(5/2)),20]

FunctionExpand[MeijerG[{{1/2,1/2},{1/2}},{{0,0,0},{}},4]/(2*Pi^(5/2))]

FullSimplify[%]

(Subscript[I, 3])==(∫_0^π)(∫_0^π)(∫_0^π)(1/(-cos(u)-cos(v)-cos(w)+3))dwdvdu

Analytic

v5.1

(1/(π^3))Integrate[(1/(3-Cos[u]-Cos[v]-Cos[w])),{u,0,π},{v,0,π},{w,0,π}]//Timing

v5.2

(1/(π^3))Integrate[(1/(3-Cos[u]-Cos[v]-Cos[w])),{u,0,π},{v,0,π},{w,0,π}]//Timing

Numerical

(1/(π^3))NIntegrate[(1/(3-Cos[u]-Cos[v]-Cos[w])),{u,0,π},{v,0,π},{w,0,π}]//Timing

(1/(π^3))NIntegrate[(1/(3-Cos[u]-Cos[v]-Cos[w])),{u,0,π},{v,0,π},{w,0,π},Method->QuasiMonteCarlo,PrecisionGoal->2,WorkingPrecision->10]//Timing

N[{
(4/(π^2))(18+12(Sqrt[2])-10(Sqrt[3])-7(Sqrt[6]))((EllipticK[((((2-(Sqrt[3]))((Sqrt[3])-(Sqrt[2]))))^2)])^2),
(18+12(Sqrt[2])-10(Sqrt[3])-7(Sqrt[6]))((EllipticTheta[3,0,Exp[-Pi (Sqrt[6])]])^4),
((Sqrt[6])/(96(π^3)))Times@@(Gamma[#/24]&/@{1,5,7,11})
},20]

Series Solution

Series[1/(1-w),{w,0,4}]

w==(Cos[x]+Cos[y]+Cos[z])/3

(∫_0^π)(∫_0^π)(∫_0^π)(1/(-cos(u)-cos(v)-cos(w)+3))\[DifferentialD]w\[DifferentialD]v\[DifferentialD]u ==(1/3)(∫_0^π)(∫_0^π)(∫_0^π)(Overscript[(∑_(k=0)), ∞])(w^k)dx dy dz
== (1/3)∫∫∫(Overscript[(∑_(k=0)), ∞])(1/(3^k))(((Cos[x]+Cos[y]+Cos[z]))^k)dx dy dz

(1/3)Sum[(1/(3^k))(((Cos[x]+Cos[y]+Cos[z]))^k),{k,0,∞}]//FullSimplify

(3/(((2π))^3))NIntegrate[(1/3)Sum[(1/(3^k))(((Cos[x]+Cos[y]+Cos[z]))^k),{k,0,∞}],{x,-Pi,0,Pi},{y,-Pi,0,Pi},{z,-Pi,0,Pi}]

FullSimplify[Integrate[Cos[x]^n1,{x,-Pi,Pi},Assumptions->n1∈Integers&&n1≥0],Assumptions->n1∈Integers&&n1≥0]

FullSimplify[Times@@Function[n1,(((((-1))^n1) (2^(1+n1)) (π^2))/((n1! )^2((Gamma[(1/2)-(n1/2)])^2)))]/@{n1,n2,n3}]

FullSimplify[((((-1))^(n1+n2+n3)) (2^(3+n1+n2+n3)) )/((((n1!))^2) (((n2!))^2) (((n3!))^2) ((Gamma[(1/2)-(n1/2)])^2) ((Gamma[(1/2)-(n2/2)])^2) ((Gamma[(1/2)-(n3/2)])^2))/.n3->k-n1-n2]

Sum[((((-1))^(n1+n2+n3)) (2^(3+n1+n2+n3)) )/((((n1!))^2) (((n2!))^2) (((n3!))^2) ((Gamma[(1/2)-(n1/2)])^2) ((Gamma[(1/2)-(n2/2)])^2) ((Gamma[(1/2)-(n3/2)])^2))/.n3->k-n1-n2,{n1,0,k},{n2,0,k-n1}]

Correct for even k but should be 0 for odd k

FullSimplify[((((-1))^k) (2^(3+k)) HypergeometricPFQ[{-(k/2),-(k/2),-(k/2)},{1,(1/2)-(k/2)},(1/4)])/((π^2) ((Gamma[(1/2)-(k/2)])^2) ((Gamma[1+(k/2)])^2) Gamma[1+k])/.k->2n,n∈Integers&&n>=0]

π^6(3/(((2π))^3))Sum[(((2n)!)/(3^(2n)))((2^(3+2 n)) HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)])/((π^2) ((Gamma[(1/2)-n])^2) ((Gamma[1+n])^2) Gamma[1+2 n]),{n,0,500}]//N//Timing

FullSimplify[(((2n)!)/(3^(2n)))((2^(3+2 n)) HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)])/((π^2) ((Gamma[(1/2)-n])^2) ((Gamma[1+n])^2) Gamma[1+2 n])]

π^6(3/(((2π))^3))(1/3)Sum[((2^(3+2 n)) (9^(-n)) HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)])/((π^2) ((Gamma[(1/2)-n])^2) ((Gamma[1+n])^2)),{n,0,∞}]

π NSum[((((4/9)))^n)((HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)])/(((Gamma[(1/2)-n])^2) ((Gamma[1+n])^2))),{n,0,100}]

π NSum[((((4/9)))^n)((HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)])/(((((((((-1))^n)(2^n))/((2n-1)!!))Sqrt[π]))^2) ((Gamma[1+n])^2))),{n,0,100}]

Sum[((((1/9)))^n)(((((2n-1)!!))^2)HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)])/ ((Gamma[1+n])^2),{n,0,200}]//N

Sum[((((1/9)))^n)((2n)!^2HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)])/ 4^n n!^2 ((n!)^2),{n,0,100}]//N

FullSimplify[((((1/9)))^n)((2n)!^2HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)])/ 4^n n!^2 ((n!)^2)]

Table[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,5}]

ListPlot[Table[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,100}],PlotJoined->True,PlotStyle->Red]

(1/3)NSum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,0,∞},NSumTerms->100,NSumExtraTerms->100,Method->Fit]//Timing

(1/3)NSum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,0,∞},NSumTerms->200,NSumExtraTerms->200,Method->Fit]//Timing

(1/3)NSum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,0,∞},NSumTerms->200,NSumExtraTerms->200,Method->Integrate]//Timing

(1/3)Sum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,0.,1000,1}]//N//Timing

(1/3)NSum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1.,(1/2)-n},(1/4)],{n,0,∞}]//N//Timing

Sums

2 (∑_k^∞)exp(π (Sqrt[6]) (-(k^2)))+1==(EllipticTheta[3, 0, (e^(-(Sqrt[6]) π))])

1+2Sum[Exp[-(k^2)π(Sqrt[6])],{k,∞}]

(∑_(n1=0)^k)(∑_(n2=0)^(k-n1))((((((-1))^(n1+n2+n3)) (2^(n1+n2+n3+3)))/((((n1!))^2) (((n2!))^2) (((n3!))^2) ((Gamma[(1/2)-(n1/2)])^2) ((Gamma[(1/2)-(n2/2)])^2) ((Gamma[(1/2)-(n3/2)])^2)))/.n3->k-n1-n2)==(((Gamma[(1/24)]) (Gamma[(5/24)]) (Gamma[(7/24)]) (Gamma[(11/24)]))/(16 (Sqrt[6]) (π^3)))

Sum[((((-1))^(n1+n2+n3)) (2^(3+n1+n2+n3)) )/((((n1!))^2) (((n2!))^2) (((n3!))^2) ((Gamma[(1/2)-(n1/2)])^2) ((Gamma[(1/2)-(n2/2)])^2) ((Gamma[(1/2)-(n3/2)])^2))/.n3->k-n1-n2,{n1,0,k},{n2,0,k-n1}]

FullSimplify[%/.k->2n,n∈Integers&&n>0]

(1/3) (∑_(n=0)^∞)((((((2 n)!))^2) (Subscript[, 3])(Subscript[F, 2])(-n,-n,-n;1,(1/2)-n;(1/4)))/((36^n) (((n!))^4)))==(((Gamma[(1/24)]) (Gamma[(5/24)]) (Gamma[(7/24)]) (Gamma[(11/24)]))/(16 (Sqrt[6]) (π^3)))

(1/3)Sum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,0,∞}]

(1/3)Sum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,0,2000}]//N

(1/3)NSum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,0,∞}]//Timing

((Sqrt[6])/(96(π^3)))Times@@Gamma[{1,5,7,11}/24]

%//N

V7

(1/3)Sum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,0,∞}]//Timing

TimeConstrained[(1/3)NSum[(((((2 n)!))^2)/((36^n)(((n!))^4)))  HypergeometricPFQ[{-n,-n,-n},{1,(1/2)-n},(1/4)],{n,0,∞}],300]//Timing

First Integral

Integrate[1/(3-Cos[x]-Cos[y]-Cos[z]),{z,-Pi,Pi},Assumptions->And@@(-Pi<#<Pi&/@{x,y})]//Timing

FullSimplify[Last[%],And@@(-Pi<#<Pi&/@{x,y})]

Second Integral

Integrate[(1/(Sqrt[(-4+Cos[x]+Cos[y]) (-2+Cos[x]+Cos[y])])),{x,-π,π},{y,-π,π}]

(3/(((2π))^2))NIntegrate[(1/(Sqrt[(-4+Cos[x]+Cos[y]) (-2+Cos[x]+Cos[y])])),{x,-π,0,π},{y,-π,0,π},SingularityDepth->20,MaxRecursion->20]

Watson

((2(Sqrt[2]))/(π^2))Integrate[(1/(Sqrt[(1+4(t^2)+3(t^4)((Sin[ψ])^2))(1+(t^2)((Sin[ψ])^2))])),{t,0,∞}]

((2(Sqrt[2]))/(π^2))Integrate[(1/(Sqrt[(1+4(t^2)+3(t^4)((Sin[ψ])^2))(1+(t^2)((Sin[ψ])^2))])),{ψ,0,π/2}]

((2(Sqrt[2]))/(π^2))NIntegrate[(1/(Sqrt[(1+4(t^2)+3(t^4)((Sin[ψ])^2))(1+(t^2)((Sin[ψ])^2))])),{t,0,∞},{ψ,0,π/2},PrecisionGoal->10]

In[2]:= Integrate[1/Sqrt[(1 + 4*t^2 + 3*t^4*Sin[ψ]^2)*(1 + t^2*Sin[ψ]^2)], {t,0,Infinity}]

Out[2]//InputForm= 
Integrate[1/Sqrt[(1 + t^2*Sin[ψ]^2)*(1 + 4*t^2 + 3*t^4*Sin[ψ]^2)], {t, 0, Infinity}]

In[3]:= Integrate[1/Sqrt[(1 + 4*t^2 + 3*t^4*Sin[ψ]^2)*(1 + t^2*Sin[ψ]^2)], {ψ,0,Pi/2}]

Out[3]//InputForm= 
Integrate[1/Sqrt[(1 + t^2*Sin[ψ]^2)*(1 + 4*t^2 + 3*t^4*Sin[ψ]^2)], {ψ, 0, Pi/2}]