Greetings to all. The title describes it well, I am writing about testing the limits of the Maple integration engine. A recent discussion at math.stackexchange.com features a family of integrals that involve the product of a power of the natural logarithm and a rational function, more precisely,
int((log(x))^n/(x^3+1), x=0..infinity);
These integrals can be evaluated recursively as described at the MSE link using a technique that generalizes to other types of rational factors. Unfortunately Maple apparently only finds a simple closed form for a few small initial values of n. The following transcript of a Maple session illustrates the problem. Mathematica was successful here. Also observe the memory allocation in the Maple session.
|\^/| Maple 18 (X86 64 LINUX)
._|\| |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2014
\ MAPLE / All rights reserved. Maple is a trademark of
<____ ____> Waterloo Maple Inc.
| Type ? for help.
> restart; read `cl.maple`;
alpha := (n, k) ->
n
-1/3 exp(1/3 I Pi + 2/3 I Pi k) (1/3 I Pi + 2/3 I Pi k)
Q := proc(n)
local res;
option remember;
if n = 0 then return 2/9*sqrt(3)*Pi end if;
res := -add(alpha(n + 1, k), k = 0 .. 2)/(n + 1) - add(
binomial(n + 1, p)*(2*I*Pi)^(n - p)*Q(p),
p = 0 .. n - 1)/(n + 1);
simplify(res)
end proc
infinity
/ n
| log(x)
VERIF := n -> | ------- dx
| 3
/ x + 1
0
> Q(6);
7 1/2
910 Pi 3
------------
6561
> VERIF(6);
memory used=3.8MB, alloc=40.3MB, time=0.18
7 1/2
9890 Pi 3 490 5 1/2
------------- + ----- Pi 3 Psi(1, 1/3)
177147 19683
490 5 1/2 10 3 1/2 2
+ ----- Pi 3 Psi(1, 2/3) + ---- Pi 3 Psi(1, 1/3)
19683 2187
20 1/2 3
+ ---- 3 Pi Psi(1, 2/3) Psi(1, 1/3)
2187
10 3 1/2 2 40 4
+ ---- Pi 3 Psi(1, 2/3) + ----- Psi(2, 2/3) Pi
2187 19683
10 1/2 3
+ ---- 3 Pi Psi(1, 1/3)
2187
10 1/2 2
+ --- Psi(1, 2/3) 3 Pi Psi(1, 1/3)
729
10 1/2 2
+ --- 3 Pi Psi(1, 1/3) Psi(1, 2/3)
729
10 1/2 3 40 4
+ ---- 3 Pi Psi(1, 2/3) - ----- Pi Psi(2, 1/3)
2187 19683
20 2 1/2
+ ---- Psi(2, 2/3) 3 Pi
6561
40 1/2
- ---- Psi(2, 2/3) 3 Psi(2, 1/3) Pi
6561
40 2
+ ---- Pi Psi(2, 2/3) Psi(1, 1/3)
2187
40 2
+ ---- Pi Psi(2, 2/3) Psi(1, 2/3)
2187
20 1/2 2
+ ---- 3 Psi(2, 1/3) Pi
6561
40 2
- ---- Pi Psi(1, 1/3) Psi(2, 1/3)
2187
40 2
- ---- Pi Psi(1, 2/3) Psi(2, 1/3)
2187
> evalf(Q(6));
725.5729634
> evalf(VERIF(6));
725.5729630
> quit
memory used=22.4MB, alloc=44.3MB, time=0.47
user@host:~/complex-logint$ math
Mathematica 10.0 for Linux x86 (64-bit)
Copyright 1988-2014 Wolfram Research, Inc.
In[1]:= Integrate[Log[z]^6/(1+z^3), {z, 0, Infinity}]
7
910 Pi
Out[1]= ------------
2187 Sqrt[3]
In[2]:= N[Out[1]]
Out[2]= 725.573
In[3]:=
user@host:~/complex-logint$
My question for you all is what the appropriate techniques would be to get Maple to at least simplify the rather involved output from the integration engine to obtain a match of the closed form from the recursive equation.
Best regards, Marko Riedel.
cl-maple.txt
