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Re: More integration/ fullsimplify bugs in Mathematica 4.1

  • To: mathgroup at smc.vnet.net
  • Subject: [mg41722] Re: More integration/ fullsimplify bugs in Mathematica 4.1
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Tue, 3 Jun 2003 07:13:04 -0400 (EDT)
  • References: <bb4vt3$49a$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Mathematica 4.0 on Windows98: No bugs found in this example.
Interesting: Simplify gives simpler results than FullSimplify.

In[53]:=
ClearAll["Global`*"]

In[54]:=
f[x_] := Integrate[1/(a*x^2 + b*x + c/4)^n, x]

### Using (simple) Simplify

In[55]:=
Simplify[f[t]]

Out[55]=
1/(a*(-1 + n))*(2^(-1 - n)*(-b + Sqrt[b^2 - a*c] - 2*a*t)*
     ((b + Sqrt[b^2 - a*c] + 2*a*t)/Sqrt[b^2 - a*c])^n*
     Hypergeometric2F1[1 - n, n, 2 - n,
      (-b + Sqrt[b^2 - a*c] - 2*a*t)/(2*Sqrt[b^2 - a*c])])/
    (c/4 + t*(b + a*t))^n

In[56]:=
Simplify[D[%, t]]

Out[56]=
(c/4 + t*(b + a*t))^(-n)

In[57]:=
Simplify[D[f[t], t]]

Out[57]=
(c/4 + t*(b + a*t))^(-n)

### Now, using FullSimplify

In[62]:=
FullSimplify[f[t]]

Out[62]=
1/a*(2^(-1 - n)*(b - Sqrt[b^2 - a*c] + 2*a*t)*
     (1 + (b + 2*a*t)/Sqrt[b^2 - a*c])^n*Gamma[1 - n]*
     Hypergeometric2F1Regularized[1 - n, n, 2 - n,
      1/2 - (b + 2*a*t)/(2*Sqrt[b^2 - a*c])])/
    (c/4 + t*(b + a*t))^n

In[63]:=
FullSimplify[D[%, t]]

Out[63]=
((-1 + n)*Gamma[1 - n]*
    (-1 + (1/2 + (b + 2*a*t)/(2*Sqrt[b^2 - a*c]))^n*
      (-1 + Gamma[2 - n])*Hypergeometric2F1Regularized[
       1 - n, n, 2 - n, 1/2 - (b + 2*a*t)/
         (2*Sqrt[b^2 - a*c])]))/(c/4 + t*(b + a*t))^n

### but still

In[65]:=
FullSimplify[D[f[t], t]]

Out[65]=
(c/4 + t*(b + a*t))^(-n)

Wolfgang

Richard Fateman wrote:

> Integrate[1/(a*x^2 + b*x + c/4)^n, x]
> FullSimplify[%]
> D[%,x]
> InputForm[FullSimplify[%]] ==>
> 
>     (4^n*(-1 + n)*Gamma[1 - n]*
>    (-1 + (1/2 + (b + 2*a*x)/(2*Sqrt[b^2 - a*c]))^n*
>      (-1 + Gamma[2 - n])*Hypergeometric2F1Regularized[
>       1 - n, n, 2 - n, 1/2 - (b + 2*a*x)/
>         (2*Sqrt[b^2 - a*c])]))/(c + 4*True*(b + a*True))^n
> 
> notice the denominator has some variable "True" in it.
> 
> Cheers.
> RJF
> 
> 


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