Bug in Mathematica 6 - Integrate - 33 - (Sinh, Cosh, Sqrt, unnecessarily arcane unsimplifiable output)
Vladimir Bondarenko
This is a bug because for a freshman level integral Mathematica,
out of the blue, returns a sophisticated output which cannot be
further simplified by any combinations of Simplify/FullSimplify
and FunctionExpand.
In this test case, it's easy for a freshman to find a substitution.
Derive 6.1 for $99 and Maple 11 calculate it instantly, too.
INT(SINH(z)/SQRT(COSH(z)^3),z,0,inf)
int(sinh(z)/sqrt(cosh(z)^3), z=0..infinity);
2
2
But Mathematica yields an eeire (from the viewpoint of a freshman)
answer, and worse still, attempts to simplify the weird stuff fail.
Integrate[Sinh[z]/Sqrt[Cosh[z]^3], {z, 0, Infinity}]
1 + 2 Sqrt[2/Pi] Gamma[5/4]^2 - (2 Sqrt[2]
Hypergeometric2F1[5/4, 3/2, 9/4, -1])/5
FullSimplify[%]
1 + 2 Sqrt[2/Pi] Gamma[5/4]^2 - (2 Sqrt[2]
Hypergeometric2F1[5/4, 3/2, 9/4, -1])/5
FunctionExpand[%]
1 + (4*Sqrt[2]*((-5*(3/Sqrt[2] - 2*(3/(2*Sqrt[2]) + (3*(-1)^(3/4
)*Beta[-1, -3/4, 1/2])/8)))/4 + 3*((-5*(-1/(2*Sqrt[2]) - ((-1)^(
3/4)*Beta[-1, 1/4, 1/2])/4))/3 + (5*(-(1/Sqrt[2]) - ((-1)^(3/4)*
Beta[-1, 1/4, 3/2])/2))/4)))/5 + 2*Sqrt[2/Pi]*Gamma[5/4]^2
etc
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
David W. Cantrell
> Hello again from the VM machine...
Vladimir, you should drastically reduce the number of groups to which you
post. I'm posting this response only to sci.math.symbolic.
> This is a bug because for a freshman level integral Mathematica,
> out of the blue, returns a sophisticated output which cannot be
> further simplified by any combinations of Simplify/FullSimplify
> and FunctionExpand.
>
> In this test case, it's easy for a freshman to find a substitution.
> Derive 6.1 for $99
Where? Sadly, according to the TI web site last month:
"As of after June 29th 2007 Texas Instruments Inc. will no longer be
selling Derive. However, TI will continue to support Derive
customers when possible. As Texas Instruments moves forward our plan
is to incorporate Derive features and compatibility into new TI
products. TI will continue to review our hardware and software
product offerings and will develop products that meet the customer and
market needs. Customers that contact 1-800-TI-Cares will receive
confirmation of the above plans if asked and any support we may be
able to provide."
> and Maple 11 calculate it instantly, too.
>
> INT(SINH(z)/SQRT(COSH(z)^3),z,0,inf)
> int(sinh(z)/sqrt(cosh(z)^3), z=0..infinity);
>
> 2
> 2
>
> But Mathematica yields an eeire (from the viewpoint of a freshman)
> answer, and worse still, attempts to simplify the weird stuff fail.
>
> Integrate[Sinh[z]/Sqrt[Cosh[z]^3], {z, 0, Infinity}]
>
> 1 + 2 Sqrt[2/Pi] Gamma[5/4]^2 - (2 Sqrt[2]
> Hypergeometric2F1[5/4, 3/2, 9/4, -1])/5
The antiderivative in Mathematica (version 5.1, at least) simplifies
nicely:
In[7]:= Integrate[Sinh[z]/Sqrt[Cosh[z]^3], z]
Out[7]= -((2*Cosh[z])/Sqrt[Cosh[z]^3])
In[8]:= FullSimplify[%, Element[z, Reals]]
Out[8]= -2*Sqrt[Sech[z]]
And of course the simple value of the definite integral can be easily
obtained from that, even by a freshman.
David
Vladimir Bondarenko
DWC> Vladimir, you should drastically reduce the number
DWC> of groups to which you post.
1) Hire me for solid money. 2) Write me, "You should ..."
I owe you nothing.
DWC> Where? [to buy Derive]
http://www.cymaths.co.uk/khxc/gbu0-catshow/Derive-6.html
Derive 6.1 Educational Single User Licence £69.33 inc VAT
Derive 6.1 Commercial Single User Licence £141.00 inc VAT
DWC> And of course the simple value of the definite integral
DWC> can be easily obtained from that, even by a freshman.
So, a fortiori, Mathematica should do this automatically. Or
at worst to be able to simplify the output to 2.
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
dimitris
Vladimir Bondarenko :
Hello Vladimir.
> This is a bug because for a freshman level integral Mathematica...
I don't see a reason why this is a bug.
Mathematica returns a correct answer!
Ok, it returns a complicated answer; I agree.
But this is not a bug. Tell is a feature.
It is because of the method used for improper integrals that appears
here this complicated result.
If we use the NL formula we simply get
In[383]:=
Integrate[Sinh[z]/Sqrt[Cosh[z]^3], z]
Limit[%, z -> Infinity] - Limit[%, z -> 0]
Out[383]=
-((2*Cosh[z])/Sqrt[Cosh[z]^3])
Out[384]=
2
Dimitris
dimitris
dimitris :
Comrade Vladimir,
You wrote
>So, a fortiori, Mathematica should do this automatically. Or
>at worst to be able to simplify the output to 2.
In this case I fully agree with you. That is, I believe this
thread has to more with a problematic behavior of Mathematica's
"simplificator" rather than Integrate itself.
Anyway...surely note the best way, but still a way
(of course this performance is not exactly what I wait for
Mathematica...)
In[482]:=
FullSimplify[FunctionExpand[Integrate[Sinh[z]/Sqrt[Cosh[z]^3], {z, 0,
Infinity}]] /.
Beta[z_, a_, b_] -> ((-(1 - z)^b)*z^(-1 + a) + (-1 + a)*Beta[z, -1
+ a, b])/(-1 + a + b)]
Out[482]=
2
David W. Cantrell
> On Jul 7, 3:57 am, David W. Cantrell <DWCantr...@sigmaxi.net> wrote:
[Newsgroups trimmed to sci.math.symbolic .]
> DWC> Vladimir, you should drastically reduce the number
> DWC> of groups to which you post.
>
> 1) Hire me for solid money.
Why? I can't see that hiring someone would encourage them follow Usenet
etiquette.
> 2) Write me, "You should ..."
That _is_ what I wrote; I didn't write "You must..."
> I owe you nothing.
Nor I you. I was merely hoping that you would be reasonable and follow
Usenet etiquette. (BTW, I suppose that requesting that you bottom-post
would also be requesting too much.)
> DWC> Where? [to buy Derive]
>
> http://www.cymaths.co.uk/khxc/gbu0-catshow/Derive-6.html
>
> Derive 6.1 Educational Single User Licence =A369.33 inc VAT
> Derive 6.1 Commercial Single User Licence =A3141.00 inc VAT
Thank you for that link. But as best I can tell, that does not help those
who live in North America or Australia.
David