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From: Vladimir Bondarenko on 22 Jul 2010 09:15
For the same integral, Wolfram Alpha returns TWO invalid answers,
see the pdfs below:

Integrate[Sin[z]^2/Sinh[z]^2, {z, 0, Infinity}]

(1/4)*(-1 + 2*Pi*Coth[Pi]) = 1.32667 (often)

-3/4 (very rarely)

Both of them are wrong as

NIntegrate[Sin[z]^2/Sinh[z]^2, {z, 0, Infinity}]

1.07667

The correct answer is

(Pi Coth[Pi] - 1)/2 = 1.07667

http://www.cybertester.com/Private/MMA7/WolframAlpha-Integrate-Bug-1a-Jul-22-2010.pdf

http://www.cybertester.com/Private/MMA7/WolframAlpha-Integrate-Bug-1b-Jul-22-2010.pdf

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http://groups.google.mk/group/comp.soft-sys.math.maple/tree/browse_frm/month/2008-03/ee443f7a8ef982b4?rnum=51&_done=/group/comp.soft-sys.math.maple/browse_frm/month/2008-03%3F#doc_1d634928d9bfae41

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If the same bugs exists through numerous software releases,
I think that is valuable public information.

It just should not happen.

-- Brad Cooper

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-- NOT FIXED BUGS ARE DANGEROUS: THEY TEND TO GET MORE SEVERE --
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Quality of CASs is our #1 care.

So our little demo continues.... Hello again from the VM machine
which hopefully soon will be used by CAS manufacturers to the
benefit of their customers.

This example demonstrates YET ANOTHER case of bad defects in the
Wolfram Research Quality Assurance process. Consider, it cannot
trap & fix efficiently MULTIPLE regression bugs the long livers.

This is also an example of a MULTIPLE regression bug when the
invalid answer is now here, now not depending on the concrete
version.

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N[ (Pi Coth[Pi] - 1)/2 ]
NIntegrate[Sin[z]^2/Sinh[z]^2, {z, 0, Infinity}]

1.07667
1.07667

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N[Integrate[Sin[z]^2/Sinh[z]^2, {z, 0, Infinity}]]

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VERSION OUTPUT RESOLUTION
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Mathematica 6.0 1.32667 <----------------------------- BUG #2

Mathematica 5.2 Correct answer (compact, 1.07667) OK

Mathematica 4.2 Unevaluated <------------------------- BUG #1

Mathematica 3.0 Correct answer (lengthy, 1.07667) OK

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Best wishes,

Vladimir Bondarenko

VM and GEMM architect
Co-founder, CEO, Mathematical Director

http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing

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like these are the way of the future."

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