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Re: Difficult integral


The problem is that Mathematica's Factor is sometimes not optimal.
One possibility to do you Integral is to use linearity and some
fiddling (probably you can guess the general form quite easily).

In[1]:= $PrePrint=InputForm;

In[2]:= i[rho_]:=Map[Factor,(ExpandAll//@Numerator[#])/Denominator[#]]&[
        Together[Map[Function[z,Integrate[z,{x,0,2Pi}]],Expand[TrigToExp[
        Sin[x]^(2 rho) Exp[I n x] ] ] ]
        ]
        ]

In[3]:= i at 4

Out[3]= 
(-40320*I*(-1 + E^(I*n*Pi))*(1 + E^(I*n*Pi)))/((-8 + n)*(-6 + n)*(-4 + n)*
  (-2 + n)*n*(2 + n)*(4 + n)*(6 + n)*(8 + n))

In[4]:= i at 10 // Timing

Out[4]= 
{32.02*Second, (-2432902008176640000*I*(-1 + E^(I*n*Pi))*(1 + E^(I*n*Pi)))/
  ((-20 + n)*(-18 + n)*(-16 + n)*(-14 + n)*(-12 + n)*(-10 + n)*(-8 + n)*
   (-6 + n)*(-4 + n)*(-2 + n)*n*(2 + n)*(4 + n)*(6 + n)*(8 + n)*(10 + n)*
   (12 + n)*(14 + n)*(16 + n)*(18 + n)*(20 + n))}

In[5]:= Timing[i at 12]

Out[5]= 
{35.87*Second, (-620448401733239439360000*I*(-1 + E^(I*n*Pi))*
   (1 + E^(I*n*Pi)))/((-24 + n)*(-22 + n)*(-20 + n)*(-18 + n)*(-16 + n)*
   (-14 + n)*(-12 + n)*(-10 + n)*(-8 + n)*(-6 + n)*(-4 + n)*(-2 + n)*n*
   (2 + n)*(4 + n)*(6 + n)*(8 + n)*(10 + n)*(12 + n)*(14 + n)*(16 + n)*
   (18 + n)*(20 + n)*(22 + n)*(24 + n))}

In[6]:= Timing[i at 14]

Out[6]= 
{51.769999999999996*Second, (-304888344611713860501504000000*I*
   (-1 + E^(I*n*Pi))*(1 + E^(I*n*Pi)))/((-28 + n)*(-26 + n)*(-24 + n)*
   (-22 + n)*(-20 + n)*(-18 + n)*(-16 + n)*(-14 + n)*(-12 + n)*(-10 + n)*
   (-8 + n)*(-6 + n)*(-4 + n)*(-2 + n)*n*(2 + n)*(4 + n)*(6 + n)*(8 + n)*
   (10 + n)*(12 + n)*(14 + n)*(16 + n)*(18 + n)*(20 + n)*(22 + n)*(24 + n)*
   (26 + n)*(28 + n))}


- - -
Rolf Mertig
Mertig Research & Consulting
http://www.mertig.com


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