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Integration of rational functions (reprise!)

• To: mathgroup at smc.vnet.net
• Subject: [mg75226] Integration of rational functions (reprise!)
• From: dimitris <dimmechan at yahoo.com>
• Date: Sat, 21 Apr 2007 23:07:10 -0400 (EDT)

Although Integrate has become undoubtfully (even) more powerful in
versions 5.X,
I think there are still some things of special attention.

Integration of rational functions is one such issue.
In particular during cases that the RootSum function is generated,
strange behavior can often be encountered. Although many threads
were based on it (i.e. RootSum) I still find intresting to point out
some of my findings.

For example

In[115]:=
g[x_] := (2*x)/((x + 1)*(x^3 + 3*x^2 + 2*x + 1))

The definite integral stays unevaluated

In[116]:=
Timing[Integrate[g[x], {x, 0, Infinity}]]

Out[116]=
{142.078*Second, Integrate[(2*x)/((1 + x)*(1 + 2*x + 3*x^2 + x^3)),
{x, 0, Infinity}]}

even though an antiderivative continuous in the integration range can
be obtained
by Integrate

In[117]:=
G[x_] = Integrate[g[x], x]

Out[117]=
2*(-Log[1 + x] + RootSum[1 + 2*#1 + 3*#1^2 + #1^3 & , (Log[x - #1] +
2*Log[x - #1]*#1 + Log[x - #1]*#1^2)/
      (2 + 6*#1 + 3*#1^2) & ])

In[118]:=
Timing[(Limit[G[x], x -> #1] & ) /@ {0, Infinity}]

Out[118]=
{191.422*Second, {2*RootSum[1 + 2*#1 + 3*#1^2 + #1^3 & , (Log[-#1] +
2*Log[-#1]*#1 + Log[-#1]*#1^2)/(2 + 6*#1 + 3*#1^2) & ], 0}}

In[119]:=
(Plus[#2 - #1] & ) @@ %[[2]]
N[%]

Out[119]=
-2*RootSum[1 + 2*#1 + 3*#1^2 + #1^3 & , (Log[-#1] + 2*Log[-#1]*#1 +
Log[-#1]*#1^2)/(2 + 6*#1 + 3*#1^2) & ]
Out[120]=
0.37121697526024766 + 0.*I

In[121]:=
(Plus[#2 - #1] & ) @@ %[[2]]
N[%]

Out[121]=
-2*RootSum[1 + 2*#1 + 3*#1^2 + #1^3 & , (Log[-#1] + 2*Log[-#1]*#1 +
Log[-#1]*#1^2)/(2 + 6*#1 + 3*#1^2) & ]
Out[122]=
0.37121697526024766 + 0.*I

In[122]:=
NIntegrate[g[x], {x, 0, Infinity}]

Out[122]=
0.3712169752602472

A somehow "opposite" behavior is also possible

E g

In[15]:=
f2[x_] := (2*x - 5)/(x^3 + 2*x^2 + 11*x + 6)

In[19]:=
F2[x_] = Integrate[f2[x], x]

Out[19]=
RootSum[6 + 11*#1 + 2*#1^2 + #1^3 & , (-5*Log[x - #1] + 2*Log[x -
#1]*#1)/(11 + 4*#1 + 3*#1^2) & ]

The definite integral can be evaluated

In[44]:=
Timing[Integrate[f2[x], {x, 0, Infinity}]]

Out[44]=
{43.687*Second, (1/3628)*(2*Log[-Root[6 + 11*#1 + 2*#1^2 + #1^3 & ,
3]]*(1556 + 771*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 3] + (189 -
6*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 1]*Root[6 + 11*#1 + 2*#1^2 + #1^3
& , 2])*
       Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 3]^2 + (33 - 9*Root[6 +
11*#1 + 2*#1^2 + #1^3 & , 1]*
         Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 2])*Root[6 + 11*#1 +
2*#1^2 + #1^3 & , 3]^3) +
    Log[-Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 1]]*(313 + 462*Root[6 +
11*#1 + 2*#1^2 + #1^3 & , 3] -
      33*(-5 + 2*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 1])*Root[6 + 11*#1
+ 2*#1^2 + #1^3 & , 3]^2 -
      3*(-5 + 2*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 1])*Root[6 + 11*#1
+ 2*#1^2 + #1^3 & , 2]^2*
       (11 + 4*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 3] + 3*Root[6 +
11*#1 + 2*#1^2 + #1^3 & , 3]^2) +
      6*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 2]*(77 + 28*Root[6 + 11*#1
+ 2*#1^2 + #1^3 & , 3] +
        (10 - 4*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 1])*Root[6 + 11*#1
+ 2*#1^2 + #1^3 & , 3]^2)) +
    Log[-Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 2]]*(-3605 + 42*Root[6 +
11*#1 + 2*#1^2 + #1^3 & , 1]*
       Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 3]*(4 + 3*Root[6 + 11*#1 +
2*#1^2 + #1^3 & , 3]) +
      9*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 1]^2*Root[6 + 11*#1 +
2*#1^2 + #1^3 & , 3]*
       (14 + 5*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 3]) - 6*Root[6 +
11*#1 + 2*#1^2 + #1^3 & , 2]*
       (77 + 3*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 1]^2*Root[6 + 11*#1
+ 2*#1^2 + #1^3 & , 3]^2) +
      3*Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 2]^2*(-55 + 22*Root[6 +
11*#1 + 2*#1^2 + #1^3 & , 3] +
        Root[6 + 11*#1 + 2*#1^2 + #1^3 & , 1]*(22 + 8*Root[6 + 11*#1 +
2*#1^2 + #1^3 & , 3]))))}

However application of the Newton-Leibniz formula fails!

In[21]:=
Limit[F2[x], x -> Infinity] - Limit[F2[x], x -> 0]

Out[21]=
Limit[RootSum[6 + 11*#1 + 2*#1^2 + #1^3 & , (-5*Log[x - #1] + 2*Log[x
- #1]*#1)/(11 + 4*#1 + 3*#1^2) & ], x -> Infinity] - RootSum[6 + 11*#1
+ 2*#1^2 + #1^3 & , (-5*Log[-#1] + 2*Log[-#1]*#1)/(11 + 4*#1 + 3*#1^2)
& ]