# Question:Investigating the limits of the Maple integration engine.

## Question:Investigating the limits of the Maple integration engine.

Maple 18

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
<____ ____>  Waterloo Maple Inc.
|       Type ? for help.
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
Mathematica 10.0 for Linux x86 (64-bit)

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]:=