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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90798] Re: Can't integrate sqrt(a+b*cos(t)+c*cos(2t))
  • From: Grischika at mail.ru
  • Date: Thu, 24 Jul 2008 04:55:03 -0400 (EDT)
  • References: <g6710s$sb6$1@smc.vnet.net>

On 23 =C9=C0=CC, 13:26, Valeri Astanoff <astan... 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}]
>
> 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

Hello.
You can try to take indefinite integral:

eq=Integrate[Sqrt[5 - 4*Cos[t] + Cos[2*t]], t]

Here Mathematica gives you an answer:

((2/5 + (4*I)/5)*Cos[t/2]^4*((2 + I)*Sqrt[1 - 2*I]*
    EllipticE[I*ArcSinh[Sqrt[1 - 2*I]*Tan[t/2]], -3/5 + (4*I)/5]*
    Sec[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]*Sec[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]*Sec[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))))/
 Sqrt[5 - 4*Cos[t] + Cos[2*t]]

Then find Limits:
Limit[eq, t ->0]

gives 0,

Limit[eq, t -> Pi]

gives

(2/5 + I/5)*Sqrt[2/5 + (4*I)/5]*
 ((1 + 2*I)*Sqrt[5]*EllipticE[-3/5 - (4*I)/5] -
  5*EllipticE[-3/5 + (4*I)/5] + (4 - 4*I)*Sqrt[5]*
   EllipticK[-3/5 - (4*I)/5] - (10 - 10*I)*EllipticK[8/5 - (4*I)/5] -
  (4 + 8*I)*EllipticPi[1/5 + (2*I)/5, -3/5 + (4*I)/5] -
  4*Sqrt[5]*EllipticPi[1 - 2*I, -3/5 - (4*I)/5])

so, the result is
N@%

-6.72287972344033 - 9.947771772989*^-15*I

The problem only with the sign of result.



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