Walking Randomly

Hello,

As our multi-years research based on a failure prediction oracle named
the VM machine reveals, unfortunately, as of Jan 20, 2009, the current
commercial computer algebra systems like Maple 12, Mathematica 7 as well
as the engineering packages like MATLAB 2008b are really crammed full
with various types of defects, from crashes to hangs to invalid, often
mathematically ridiculous answers.

For your general case, Maple 12 returns sophisticated output

limit(1/2*(-k*((1+I*x)/(1+x^2))^(1/2)+(-(-1-x^2+k^2+I*k^2*x)/(1+x^2))^(1/2)*((1+x^2-k^2+I*k^2*x)/(1+x^2))^(1/2)*EllipticF(k*((1+I*x)/(1+x^2))^(1/2),(-(-1+x^2+2*I*x)/(1+x^2))^(1/2))+2*PIECEWISE([(-signum(-1+I*x))^(1/2)*infinity*(-(-I*signum((1-I*x)^(1/2)*x))^(1/2)+(I*signum((1-I*x)^(1/2)*x))^(1/2))/(I*signum((1-I*x)^(1/2)*x))^(1/2)/(-I*signum((1-I*x)^(1/2)*x))^(1/2), 0 < (1-I*x)^(1/2)],[0, otherwise])*((1+I*x)/(1+x^2))^(1/2)*(k^4-2*k^2+1+x^2)^(1/2)+2*PIECEWISE([signum(1+I*x)^(1/2)*infinity*(-(I*signum((1+I*x)^(1/2)*x))^(1/2)+(-I*signum((1+I*x)^(1/2)*x))^(1/2))/(-I*signum((1+I*x)^(1/2)*x))^(1/2)/(I*signum((1+I*x)^(1/2)*x))^(1/2), 0 < (1+I*x)^(1/2)],[0, otherwise])*((1+I*x)/(1+x^2))^(1/2)*(k^4-2*k^2+1+x^2)^(1/2)+2*PIECEWISE([(-signum(-1+I*x))^(1/2)*infinity*(-(-I*signum((1-I*x)^(1/2)*x))^(1/2)+(I*signum((1-I*x)^(1/2)*x))^(1/2))/(I*signum((1-I*x)^(1/2)*x))^(1/2)/(-I*signum((1-I*x)^(1/2)*x))^(1/2), 0 < -(1-I*x)^(1/2)],[0, otherwise])*((1+I*x)/(1+x^2))^(1/2)*(k^4-2*k^2+1+x^2)^(1/2)+2*PIECEWISE([signum(1+I*x)^(1/2)*infinity*(-(I*signum((1+I*x)^(1/2)*x))^(1/2)+(-I*signum((1+I*x)^(1/2)*x))^(1/2))/(-I*signum((1+I*x)^(1/2)*x))^(1/2)/(I*signum((1+I*x)^(1/2)*x))^(1/2), 0 int(z^2*(z^2 – 1)/((z^2 – 1)^2 + 4)^(3/2), z= 0..infinity);

1/10*5^(3/4)*EllipticK(1/10*(50+10*5^(1/2))^(1/2))

> evalf(%);

.7060937855

> evalf(Int(z^2*(z^2 – 1)/((z^2 – 1)^2 + 4)^(3/2), z=0..infinity));

.7060937855

Cheers,

Vladimir Bondarenko
CEO, Mathematical Director

Cyber Tester Ltd.
http://www.cybertester.com

phone: +380-953-866-894
fax: +380-652-668-356

3/4 Zadorozhny Str, Simferopol
Crimea 95047, Ukraine

Maple 12 computes both symbolic and numeric integrals well.

> t:=int(k^2* (k^2 – 1)/((k^2 – 1)^2 + x^2)^(3/2),k=0..infinity) assuming x>0; # compute the integral symbolically

t := 1/2*1/(1+x^2)^(1/4)*EllipticK(1/2*2^(1/2)*(((1+x^2)^(1/2)+1)/(1+x^2)^(1/2))^(1/2))

> evalf(subs(x=2,t)); # substitute x=2 in symbolic result

.7060937851

> int(k^2* (k^2 – 1)/((k^2 – 1)^2 + 2^2)^(3/2),k=0.0..infinity); # compute integral numerically at x=2

.7060937855

“@Vladimir – Thanks also but I wonder why the result you get is so large? Did you include the assumption that x>0?”

Oooops, woe unto me, there was a typo. My input was “assuming k>0” while
it should be “assuming x>0”.

I confirm the result by VolMike.

In the given case Maple 12 beats Mathematica 7.

Why my typo occurred – and I did not spot it?

Maybe, it’s because of the following. To tell the truth, by and large, as
long as it is about integration, I got accustomed to see that Mathematica
7 beats Maple 12.

But in this case, Maplesoft got even Wolfram Research :)

Not surprisingly, Mable-based Matlab 7.02 gives the same (correct) result as Maple 12, while mupad based Matlab R2008b returns “Explicit integral could not be found.”

The simple way to avoid (decrease the probability of) such errors is comparing results in CAS’s based on different symbolic (numeric) engines.