@vv

Greetings once again. The goal being to improve the Maple integrator as much as possible and make it the best it can be as deserved by a first class computer algebra system I would like to present another application of my recursive method from the MSE post. This time we look to evaluate

int((log(x))^n/(x^3-2*x+4), x=0..infinity);

The following Maple session shows differentiating with respect to a parameter not working and Maple giving up on a closed form. The rest speaks for itself. Please do observe that I present this material to encourage an implementation of the recursive method or equivalent for the Maple integration engine.

|\^/| 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.
> int(x^a/(x^3+1), x=0..infinity);
1/3 Pi csc(1/3 Pi a + 1/3 Pi)
> int(x^a/(x^3-2*x+4), x=0..infinity);
infinity
/ a
| x
| ------------ dx
| 3
/ x - 2 x + 4
0
> restart; read `cl2.maple`;
alpha_sum := proc(n)
local poles;
poles := [[1 + I, 1/2*log(2) + 1/4*I*Pi],
[1 - I, 1/2*log(2) + 7/4*I*Pi], [-2, log(2) + Pi*I]
];
add(residue(1/(x^3 - 2*x + 4), x = p[1])*p[2]^n,
p in poles)
end proc
Q := proc(n)
local res;
option remember;
if n = 0 then return
simplify(int(1/(x^3 - 2*x + 4), x = 0 .. infinity))
end if;
res := -alpha_sum(n + 1)/(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 - 2 x + 4
0
> Q(4);
memory used=30.4MB, alloc=40.3MB, time=0.66
139 3 2 4897 4 4
- ---- ln(2) Pi - ----- ln(2) Pi + 9/640 ln(2) Pi
1920 76800
5
63 2 3 357 Pi 31 5
+ ---- ln(2) Pi + ------- - ---- ln(2)
1280 10240 1600
> evalf(%);
6.866005720
> VERIF(4);
infinity
/ 4
| ln(x)
| ------------ dx
| 3
/ x - 2 x + 4
0
> evalf(%);
6.866005710
> diff(int(x^a/(x^3-2*x+4), x=0..infinity), a$4);
memory used=65.3MB, alloc=78.3MB, time=1.49
memory used=129.8MB, alloc=86.3MB, time=2.67
infinity
/ a 4
| x ln(x)
| ------------ dx
| 3
/ x - 2 x + 4
0
user@host:~/complex-logint$ math
Mathematica 10.0 for Linux x86 (64-bit)
Copyright 1988-2014 Wolfram Research, Inc.
In[1]:= Integrate[Log[z]^4/(z^3-2*z+4), {z, 0, Infinity}]
5 4 3 2
Out[1]= (5355 Pi - 9794 Pi Log[2] + 7560 Pi Log[2] -
2 3 4 5
> 11120 Pi Log[2] + 2160 Pi Log[2] - 2976 Log[2] ) /
> 153600
In[2]:= Expand[Out[1]]
5 4 3 2
357 Pi 4897 Pi Log[2] 63 Pi Log[2]
Out[2]= ------- - --------------- + -------------- -
10240 76800 1280
2 3 4 5
139 Pi Log[2] 9 Pi Log[2] 31 Log[2]
> --------------- + ------------ - ----------
1920 640 1600
In[3]:= N[Out[1]]
Out[3]= 6.86601
In[4]:=
user@host:~/complex-logint$

Best regards, Marko Riedel

cl2-maple.txt