Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
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From: Valeri Astanoff on 23 Jul 2008 06:26 Good day, Neither Mathematica 6 nor anyone here can integrate this: In[1]:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] Out[1]= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] In[2]:= NIntegrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] Out[2]= 6.72288 I know the exact result: In[3]:= (1/5^(3/4))*(Sqrt[2]*(10*EllipticE[(1/10)*(5 - Sqrt[5])] - 10*EllipticK[(1/10)*(5 - Sqrt[5])] + (5 + 3*Sqrt[5])* EllipticPi[(1/10)*(5 - 3*Sqrt[5]), (1/10)*(5 - Sqrt[5])]))//N Out[3]= 6.72288 but I would like to prove it. Thanks in advance to the samaritan experts... V.Astanoff
From: Alois Steindl on 24 Jul 2008 04:50 Hello, If I ask Mathematica 6.0.2.1 to Integrate[Simplify[Sqrt[5 - 4*Cos[t] + Cos[2*t]]], {t, 0, Pi}] it gives me 1/5 Sqrt[ 2 + 4 I] (-5 I EllipticE[-(3/5) - (4 I)/5] + (2 + I) Sqrt[5] EllipticE[-(3/5) + (4 I)/5] - (12 - 4 I) EllipticK[-(3/5) - ( 4 I)/5] + (6 - 2 I) Sqrt[5] EllipticK[8/5 - (4 I)/5] + 4 I Sqrt[5] EllipticPi[ 1/5 + (2 I)/5, -(3/5) + (4 I)/5] + (8 + 4 I) EllipticPi[ 1 - 2 I, -(3/5) - (4 I)/5]) Taking N[%] gives 6.72288+ 6.52169*10^-15 I I am wondering, why the Simplify helps, because Simplify[Sqrt[5 - 4*Cos[t] + Cos[2*t]]] just returns Sqrt[5 - 4 Cos[t] + Cos[2 t]]. Best wishes Alois -- Alois Steindl, Tel.: +43 (1) 58801 / 32558 Inst. for Mechanics and Mechatronics Fax.: +43 (1) 58801 / 32598 Vienna University of Technology, A-1040 Wiedner Hauptstr. 8-10
From: Grischika on 24 Jul 2008 04:51 On 23 =C9=C0=CC, 13:26, Valeri Astanoff <astan...(a)gmail.com> wrote: > Good day, > > Neither Mathematica 6 nor anyone here can integrate this: > > In[1]:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] > Out[1]= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] > > In[2]:= NIntegrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] > Out[2]= 6.72288 > > I know the exact result: > > In[3]:= =9A(1/5^(3/4))*(Sqrt[2]*(10*EllipticE[(1/10)*(5 - Sqrt[5])] - > =9A =9A =9A =9A 10*EllipticK[(1/10)*(5 - Sqrt[5])] + (5 + 3*Sqrt[5])* > =9A =9A =9A =9A EllipticPi[(1/10)*(5 - 3*Sqrt[5]), (1/10)*(5 - Sqrt[5])])= )//N > Out[3]= 6.72288 > > but I would like to prove it. > > Thanks in advance to the samaritan experts... > > V.Astanoff Or even beter eq = Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi/2}] + Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, Pi/2, Pi}]; FullSimplify[eq] Out: (Sqrt[2 + 4*I]*((-5*I)*EllipticE[-3/5 - (4*I)/5] + (2 + I)*Sqrt[5]*EllipticE[-3/5 + (4*I)/5] - (12 - 4*I)*EllipticK[-3/5 - (4*I)/5] + (6 - 2*I)*Sqrt[5]* EllipticK[8/5 - (4*I)/5] + (4*I)*Sqrt[5]*EllipticPi[1/5 + (2*I)/ 5, -3/5 + (4*I)/5] + (8 + 4*I)*EllipticPi[1 - 2*I, -3/5 - (4*I)/ 5]))/ 5 In[]=N@% Out: 6.72288+ 1.05693*10^-14* I Moreover, you can replace Cos[2t]->Cos[t]^2-Sin[t]^2 so Integrate[ Sqrt[5 - 4*Cos[t] + Cos[2*t]] /. Cos[2 t] -> Cos[t]^2 - Sin[t]^2, {t, 0, Pi}]; gives result as above 6.72288+ 8.76799*10^-15 I And the last solution is may be the best one: Integrate[ TrigExpand/@Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t,0, Pi}];
From: Kevin J. McCann on 24 Jul 2008 04:52 I actually get: 1/5 Sqrt[2+4 I] (-5 I EllipticE[-(3/5)-(4 I)/5]+(2+I) Sqrt[5] EllipticE[-(3/5)+(4 I)/5]-(12-4 I) EllipticK[-(3/5)-(4 I)/5]+(6-2 I) Sqrt[5] EllipticK[8/5-(4 I)/5]+4 I Sqrt[5] EllipticPi[1/5+(2 I)/5,-(3/5)+(4 I)/5]+(8+4 I) EllipticPi[1-2 I,-(3/5)-(4 I)/5]) which evaluates to your numerical answer below, Valeri Astanoff wrote: > Good day, > > Neither Mathematica 6 nor anyone here can integrate this: > > In[1]:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] > Out[1]= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] > > In[2]:= NIntegrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] > Out[2]= 6.72288 > > I know the exact result: > > In[3]:= (1/5^(3/4))*(Sqrt[2]*(10*EllipticE[(1/10)*(5 - Sqrt[5])] - > 10*EllipticK[(1/10)*(5 - Sqrt[5])] + (5 + 3*Sqrt[5])* > EllipticPi[(1/10)*(5 - 3*Sqrt[5]), (1/10)*(5 - Sqrt[5])]))//N > Out[3]= 6.72288 > > but I would like to prove it. > > Thanks in advance to the samaritan experts... > > > V.Astanoff >
From: David W.Cantrell on 24 Jul 2008 04:53
Valeri Astanoff <astanoff(a)gmail.com> wrote: > Good day, > > Neither Mathematica 6 nor anyone here can integrate this: > > In[1]:= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] > Out[1]= Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] Actually, we can use Mathematica 6 to integrate that. In[9]:= indef = Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], t] Out[9]= ((2/5 + (4*I)/5)*Sqrt[5 - 4*Cos[t] + Cos[2*t]]*((2 + I)*Sqrt[1 - 2*I]* EllipticE[I*ArcSinh[Sqrt[1 - 2*I]*Tan[t/2]], -(3/5) + (4*I)/5]*(1 + Tan[t/2]^2)*Sqrt[1 + (1 - 2*I)*Tan[t/2]^2]*Sqrt[1 + (1 + 2*I)*Tan[t/2]^2] - I*((6 - 2*I)*Sqrt[1 - 2*I]*EllipticF[I*ArcSinh[Sqrt[1 - 2*I]*Tan[t/2]], -(3/5) + (4*I)/5]*(1 + Tan[t/2]^2)*Sqrt[1 + (1 - 2*I)*Tan[t/2]^2]*Sqrt[1 + (1 + 2*I)*Tan[t/2]^2] - 4*Sqrt[1 - 2*I]*EllipticPi[1/5 + (2*I)/5, I*ArcSinh[Sqrt[1 - 2*I]*Tan[t/2]], -(3/5) + (4*I)/5]*(1 + Tan[t/2]^2)*Sqrt[1 + (1 - 2*I)*Tan[t/2]^2]*Sqrt[1 + (1 + 2*I)*Tan[t/2]^2] + (2 + I)*(Tan[t/2] + 2*Tan[t/2]^3 + 5*Tan[t/2]^5))))/((1 + Cos[t])*Sqrt[(5 - 4*Cos[t] + Cos[2*t])/(1 + Cos[t])^2]*(1 + Tan[t/2]^2)*Sqrt[2 + 4*Tan[t/2]^2 + 10*Tan[t/2]^4]) In[10]:= FullSimplify[ Limit[indef, t -> Pi, Direction -> 1] - (indef /. t -> 0)] Out[10]= (1/5)*Sqrt[2 + 4*I]*(-5*I*EllipticE[-(3/5) - (4*I)/5] + (2 + I)*Sqrt[5]*EllipticE[-(3/5) + (4*I)/5] - (12 - 4*I)*EllipticK[-(3/5) - (4*I)/5] + (6 - 2*I)*Sqrt[5]*EllipticK[8/5 - (4*I)/5] + 4*I*Sqrt[5]*EllipticPi[1/5 + (2*I)/5, -(3/5) + (4*I)/5] + (8 + 4*I)*EllipticPi[1 - 2*I, -(3/5) - (4*I)/5]) In[11]:= N[%] Out[11]= 6.722879723440325 + 1.0534455252564358*^-14*I Of course I readily agree that Out[10] is not as nice in appearance as your In[3] below. (Nobody who works much with Mathematica and elliptic integrals would be surprised by that.) Nonetheless, Out[10] is a correct answer. Best regards, David W. Cantrell > In[2]:= NIntegrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], {t, 0, Pi}] > Out[2]= 6.72288 > > I know the exact result: > > In[3]:= (1/5^(3/4))*(Sqrt[2]*(10*EllipticE[(1/10)*(5 - Sqrt[5])] - > 10*EllipticK[(1/10)*(5 - Sqrt[5])] + (5 + 3*Sqrt[5])* > EllipticPi[(1/10)*(5 - 3*Sqrt[5]), (1/10)*(5 - Sqrt[5])]))//N > Out[3]= 6.72288 > > but I would like to prove it. > > Thanks in advance to the samaritan experts... > > V.Astanoff |