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Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90794] Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
  • From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Thu, 24 Jul 2008 04:54:18 -0400 (EDT)
  • References: <g6710s$sb6$1@smc.vnet.net>

Valeri Astanoff <astanoff at 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


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