rotten news relay - sci.math.symbolic - Re: An interesting rational integral in Mathematica

Subject: Re: An interesting rational integral in Mathematica
From: Sam Blake
Newsgroups: sci.math.symbolic
Date: Sat, 13 Jan 2024 02:56 UTC
References: 1

X-Received: by 2002:a05:6214:21ae:b0:680:dce8:a791 with SMTP id t14-20020a05621421ae00b00680dce8a791mr289201qvc.4.1705114597649;
Fri, 12 Jan 2024 18:56:37 -0800 (PST)
X-Received: by 2002:a05:6214:e4f:b0:67f:935c:3b24 with SMTP id
o15-20020a0562140e4f00b0067f935c3b24mr186860qvc.7.1705114597415; Fri, 12 Jan
2024 18:56:37 -0800 (PST)
Path: eternal-september.org!news.eternal-september.org!feeder3.eternal-september.org!news.chmurka.net!usenet.blueworldhosting.com!diablo1.usenet.blueworldhosting.com!peer02.iad!feed-me.highwinds-media.com!news.highwinds-media.com!news-out.google.com!nntp.google.com!postnews.google.com!google-groups.googlegroups.com!not-for-mail
Newsgroups: sci.math.symbolic
Date: Fri, 12 Jan 2024 18:56:37 -0800 (PST)
In-Reply-To: <b3d304f9-640f-45b2-9cf6-24e7e2853753n@googlegroups.com>
Injection-Info: google-groups.googlegroups.com; posting-host=144.130.168.222; posting-account=wlWA4gkAAABfPzIzZZgzF-hX5pK428zY
NNTP-Posting-Host: 144.130.168.222
References: <b3d304f9-640f-45b2-9cf6-24e7e2853753n@googlegroups.com>
User-Agent: G2/1.0
MIME-Version: 1.0
Message-ID: <70c96453-8dd1-420a-b03c-3be0718b1182n@googlegroups.com>
Subject: Re: An interesting rational integral in Mathematica
From: samuel.thomas.blake@gmail.com (Sam Blake)
Injection-Date: Sat, 13 Jan 2024 02:56:37 +0000
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
X-Received-Bytes: 3311

View all headers

On Saturday, January 13, 2024 at 1:47:35 PM UTC+11, Sam Blake wrote:
> Some of you may find this result interesting...
>
> In[1047]:= $Version
>
> Out[1047]= "13.2.0 for Mac OS X x86 (64-bit) (November 18, 2022)"
>
> In[1048]:= Integrate[(2 - Sqrt[7] x^2 + 3 x^4)/(2 + Sqrt[7] x^2 - x^4)^2, x]
>
> Out[1048]= 1/60 (30 x AppellF1[1/2, 2, 2, 3/2, (2 x^2)/(Sqrt[7] + Sqrt[15]), (2 x^2)/(Sqrt[7] - Sqrt[15])] -
> 5 Sqrt[7] x^3 AppellF1[3/2, 2, 2, 5/2, (2 x^2)/(Sqrt[7] + Sqrt[15]), (2 x^2)/(Sqrt[7] - Sqrt[15])] +
> 9 x^5 AppellF1[5/2, 2, 2, 7/2, (2 x^2)/(Sqrt[7] + Sqrt[15]), (2 x^2)/(Sqrt[7] - Sqrt[15])])
>
> Cheers,
>
> Sam

.... and here's an example where Rubi outperforms both Mathematica and my implementation of Mack's linear Hermite reduction (as given in Symbolic Integration I by Manuel Bronstein)

In[1151]:= (2 - 9 x^4 + 2 Sqrt[2] x^4 - 12 x^6 - 3 x^8)/(Sqrt[2] - 3 x^2 - x^4)^3;
HermiteReduce[%, x]
Integrate[%%, x]
Int[%%%, x]

Out[1152]= {-((37549921350591017222596 (102743992 x +
72655641 Sqrt[2] x))/((-8 + 9 Sqrt[2]) (-113 +
72 Sqrt[2])^4 (2200 + 1593 Sqrt[2])^2 (-32544 +
23137 Sqrt[2]) (3352883803641 +
2370846461804 Sqrt[2]) (-3186 - 2200 Sqrt[2] + 6600 x^2 +
4779 Sqrt[2] x^2 + 2200 x^4 + 1593 Sqrt[2] x^4))), 0}

Out[1153]= -(1/(
105 (-2 + 3 Sqrt[2] x^2 + Sqrt[2] x^4)^3))(Sqrt[2] - 3 x^2 -
x^4)^3 (210 x AppellF1[1/2, 3, 3, 3/2, (
2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
3 + Sqrt[9 + 4 Sqrt[2]]))] +
21 (-9 + 2 Sqrt[2]) x^5 AppellF1[5/2, 3, 3, 7/2, (
2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
3 + Sqrt[9 + 4 Sqrt[2]]))] -
5 (36 x^7 AppellF1[7/2, 3, 3, 9/2, (
2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
3 + Sqrt[9 + 4 Sqrt[2]]))] +
7 x^9 AppellF1[9/2, 3, 3, 11/2, (
2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
3 + Sqrt[9 + 4 Sqrt[2]]))]))

Out[1154]= x/(Sqrt[2] - 3 x^2 - x^4)

Subject: Re: An interesting rational integral in Mathematica
From: Sam Blake
Newsgroups: sci.math.symbolic
Date: Sat, 13 Jan 2024 06:28 UTC
References: 1 2

X-Received: by 2002:a05:6214:5085:b0:680:b25b:41d with SMTP id kk5-20020a056214508500b00680b25b041dmr22770qvb.5.1705127303563;
Fri, 12 Jan 2024 22:28:23 -0800 (PST)
X-Received: by 2002:a05:6214:20ac:b0:681:315b:255d with SMTP id
12-20020a05621420ac00b00681315b255dmr123818qvd.8.1705127303327; Fri, 12 Jan
2024 22:28:23 -0800 (PST)
Path: eternal-september.org!news.eternal-september.org!feeder3.eternal-september.org!panix!usenet.blueworldhosting.com!diablo1.usenet.blueworldhosting.com!peer03.iad!feed-me.highwinds-media.com!news.highwinds-media.com!news-out.google.com!nntp.google.com!postnews.google.com!google-groups.googlegroups.com!not-for-mail
Newsgroups: sci.math.symbolic
Date: Fri, 12 Jan 2024 22:28:22 -0800 (PST)
In-Reply-To: <70c96453-8dd1-420a-b03c-3be0718b1182n@googlegroups.com>
Injection-Info: google-groups.googlegroups.com; posting-host=144.130.168.222; posting-account=wlWA4gkAAABfPzIzZZgzF-hX5pK428zY
NNTP-Posting-Host: 144.130.168.222
References: <b3d304f9-640f-45b2-9cf6-24e7e2853753n@googlegroups.com> <70c96453-8dd1-420a-b03c-3be0718b1182n@googlegroups.com>
User-Agent: G2/1.0
MIME-Version: 1.0
Message-ID: <c4a7b82c-90af-43f3-82bc-f2414c5583a0n@googlegroups.com>
Subject: Re: An interesting rational integral in Mathematica
From: samuel.thomas.blake@gmail.com (Sam Blake)
Injection-Date: Sat, 13 Jan 2024 06:28:23 +0000
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
X-Received-Bytes: 4799

View all headers

On Saturday, January 13, 2024 at 1:56:38 PM UTC+11, Sam Blake wrote:
> On Saturday, January 13, 2024 at 1:47:35 PM UTC+11, Sam Blake wrote:
> > Some of you may find this result interesting...
> >
> > In[1047]:= $Version
> >
> > Out[1047]= "13.2.0 for Mac OS X x86 (64-bit) (November 18, 2022)"
> >
> > In[1048]:= Integrate[(2 - Sqrt[7] x^2 + 3 x^4)/(2 + Sqrt[7] x^2 - x^4)^2, x]
> >
> > Out[1048]= 1/60 (30 x AppellF1[1/2, 2, 2, 3/2, (2 x^2)/(Sqrt[7] + Sqrt[15]), (2 x^2)/(Sqrt[7] - Sqrt[15])] -
> > 5 Sqrt[7] x^3 AppellF1[3/2, 2, 2, 5/2, (2 x^2)/(Sqrt[7] + Sqrt[15]), (2 x^2)/(Sqrt[7] - Sqrt[15])] +
> > 9 x^5 AppellF1[5/2, 2, 2, 7/2, (2 x^2)/(Sqrt[7] + Sqrt[15]), (2 x^2)/(Sqrt[7] - Sqrt[15])])
> >
> > Cheers,
> >
> > Sam
> ... and here's an example where Rubi outperforms both Mathematica and my implementation of Mack's linear Hermite reduction (as given in Symbolic Integration I by Manuel Bronstein)
>
> In[1151]:= (2 - 9 x^4 + 2 Sqrt[2] x^4 - 12 x^6 - 3 x^8)/(Sqrt[2] - 3 x^2 - x^4)^3;
> HermiteReduce[%, x]
> Integrate[%%, x]
> Int[%%%, x]
>
> Out[1152]= {-((37549921350591017222596 (102743992 x +
> 72655641 Sqrt[2] x))/((-8 + 9 Sqrt[2]) (-113 +
> 72 Sqrt[2])^4 (2200 + 1593 Sqrt[2])^2 (-32544 +
> 23137 Sqrt[2]) (3352883803641 +
> 2370846461804 Sqrt[2]) (-3186 - 2200 Sqrt[2] + 6600 x^2 +
> 4779 Sqrt[2] x^2 + 2200 x^4 + 1593 Sqrt[2] x^4))), 0}
>
> Out[1153]= -(1/(
> 105 (-2 + 3 Sqrt[2] x^2 + Sqrt[2] x^4)^3))(Sqrt[2] - 3 x^2 -
> x^4)^3 (210 x AppellF1[1/2, 3, 3, 3/2, (
> 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> 3 + Sqrt[9 + 4 Sqrt[2]]))] +
> 21 (-9 + 2 Sqrt[2]) x^5 AppellF1[5/2, 3, 3, 7/2, (
> 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> 3 + Sqrt[9 + 4 Sqrt[2]]))] -
> 5 (36 x^7 AppellF1[7/2, 3, 3, 9/2, (
> 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> 3 + Sqrt[9 + 4 Sqrt[2]]))] +
> 7 x^9 AppellF1[9/2, 3, 3, 11/2, (
> 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> 3 + Sqrt[9 + 4 Sqrt[2]]))]))
>
> Out[1154]= x/(Sqrt[2] - 3 x^2 - x^4)

For fairness, Rubi makes a mess of this one

In[1578]:= Int[(-2 - 2*Sqrt[3]*x - 3*x^2 + 4*Sqrt[3]*x^3 + 7*x^4 + 6*Sqrt[3]*x^5 + 10*x^6 + 2*Sqrt[3]*x^7 + 6*x^8 + x^10)/(1 + 3*x^2 + Sqrt[3]*x^3 + 3*x^4 + x^6)^2, x]

Out[1578]= -((3*3^(1/6) + 4*Sqrt[3] -
5*3^(5/6) + (3 - 4*3^(2/3))*
x)/(3*(3 - 4*3^(2/3))*(1 + 3^(1/6)*x + x^2))) -
(3^(1/3)*(6*I - 3*(-1)^(2/3)*3^(1/6) + 2*Sqrt[3] +
5*3^(5/6) + (-1)^(1/3)*(3 + 3^(2/3)*(2 + 2*I*Sqrt[3]))*x))/
((1 + (-1)^(1/3))^4*(4 - (-1)^(2/3)*3^(1/3))*(1 - (-1)^(1/
3)*3^(1/6)*x + x^2)) -
(2*(-1)^(1/3)*(3*(-1)^(2/3)*3^(1/6) + 4*Sqrt[3] +
5*(-1)^(1/3)*3^(5/6) + (3 - 4*(-3)^(2/3))*x))/
(3*3^(2/3)*(8 + 3^(1/3) + I*3^(5/6))*(1 + (-1)^(2/3)*3^(1/6)*x +
x^2)) +
(2*ArcTan[((-1)^(1/3)*3^(1/6) - 2*x)/
Sqrt[4 - (-1)^(2/3)*3^(1/3)]])/(3*
Sqrt[(1/2)*(8 + 3^(1/3) - I*3^(5/6))]) -
(2*ArcTan[((-1)^(2/3)*3^(1/6) + 2*x)/Sqrt[4 + (-3)^(1/3)]])/(3*
Sqrt[(1/2)*(8 + 3^(1/3) + I*3^(5/6))]) +
(2*I*ArcTanh[(3^(1/6)*(I - Sqrt[3]) - 4*I*x)/
Sqrt[2*(8 + 3^(1/3) - I*3^(5/6))]])/
(3*Sqrt[(1/2)*(8 + 3^(1/3) - I*3^(5/6))]) +
(2*I*ArcTanh[(3^(1/6)*(I + Sqrt[3]) - 4*I*x)/
Sqrt[2*(8 + 3^(1/3) + I*3^(5/6))]])/
(3*Sqrt[(1/2)*(8 + 3^(1/3) + I*3^(5/6))])

Subject: Re: An interesting rational integral in Mathematica
From: clicliclic@freenet.d
Newsgroups: sci.math.symbolic
Organization: Killfiles, Unlimited
Date: Sat, 13 Jan 2024 09:29 UTC
References: 1 2

Path: eternal-september.org!news.eternal-september.org!feeder3.eternal-september.org!paganini.bofh.team!news.killfile.org!.POSTED.port-212-4-179-246.dynamic.as20676.net!not-for-mail
From: nobody@nowhere.invalid (clicliclic@freenet.de)
Newsgroups: sci.math.symbolic
Subject: Re: An interesting rational integral in Mathematica
Date: Sat, 13 Jan 2024 10:29:26 +0100
Organization: Killfiles, Unlimited
Message-ID: <65A257F6.848BF71F@nowhere.invalid>
References: <b3d304f9-640f-45b2-9cf6-24e7e2853753n@googlegroups.com> <70c96453-8dd1-420a-b03c-3be0718b1182n@googlegroups.com>
Mime-Version: 1.0
Content-Type: text/plain; charset=us-ascii
Content-Transfer-Encoding: 7bit
Injection-Info: flea.killfile.org; posting-host="port-212-4-179-246.dynamic.as20676.net:212.4.179.246";
logging-data="276067"; mail-complaints-to="news@news.killfile.org"
X-Mailer: Mozilla 4.75 [de] (Win98; U)
X-Accept-Language: de

View all headers

Sam Blake schrieb:
>
> ... and here's an example where Rubi outperforms both Mathematica and
> my implementation of Mack's linear Hermite reduction (as given in
> Symbolic Integration I by Manuel Bronstein)
>
> In[1151]:= (2 - 9 x^4 + 2 Sqrt[2] x^4 - 12 x^6 - 3 x^8)/(Sqrt[2] - 3 x^2 - x^4)^3;
> HermiteReduce[%, x]
> Integrate[%%, x]
> Int[%%%, x]
>
> Out[1152]= {-((37549921350591017222596 (102743992 x +
> 72655641 Sqrt[2] x))/((-8 + 9 Sqrt[2]) (-113 +
> 72 Sqrt[2])^4 (2200 + 1593 Sqrt[2])^2 (-32544 +
> 23137 Sqrt[2]) (3352883803641 +
> 2370846461804 Sqrt[2]) (-3186 - 2200 Sqrt[2] + 6600 x^2 +
> 4779 Sqrt[2] x^2 + 2200 x^4 + 1593 Sqrt[2] x^4))), 0}
>
> Out[1153]= -(1/(
> 105 (-2 + 3 Sqrt[2] x^2 + Sqrt[2] x^4)^3))(Sqrt[2] - 3 x^2 -
> x^4)^3 (210 x AppellF1[1/2, 3, 3, 3/2, (
> 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> 3 + Sqrt[9 + 4 Sqrt[2]]))] +
> 21 (-9 + 2 Sqrt[2]) x^5 AppellF1[5/2, 3, 3, 7/2, (
> 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> 3 + Sqrt[9 + 4 Sqrt[2]]))] -
> 5 (36 x^7 AppellF1[7/2, 3, 3, 9/2, (
> 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> 3 + Sqrt[9 + 4 Sqrt[2]]))] +
> 7 x^9 AppellF1[9/2, 3, 3, 11/2, (
> 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> 3 + Sqrt[9 + 4 Sqrt[2]]))]))
>
> Out[1154]= x/(Sqrt[2] - 3 x^2 - x^4)
>

.... with the same result being obtained by Derive 6.10:

INT((2 - 9*x^4 + 2*SQRT(2)*x^4 - 12*x^6 - 3*x^8)
/(SQRT(2) - 3*x^2 - x^4)^3, x)

- x/(x^4 + 3*x^2 - SQRT(2))

Martin.

Path: eternal-september.org!news.eternal-september.org!feeder3.eternal-september.org!paganini.bofh.team!news.killfile.org!.POSTED.port-212-4-179-246.dynamic.as20676.net!not-for-mail
From: nobody@nowhere.invalid (clicliclic@freenet.de)
Newsgroups: sci.math.symbolic
Subject: Re: An interesting rational integral in Mathematica
Date: Sat, 13 Jan 2024 10:29:31 +0100
Organization: Killfiles, Unlimited
Message-ID: <65A257FB.690DA194@nowhere.invalid>
References: <b3d304f9-640f-45b2-9cf6-24e7e2853753n@googlegroups.com> <70c96453-8dd1-420a-b03c-3be0718b1182n@googlegroups.com> <c4a7b82c-90af-43f3-82bc-f2414c5583a0n@googlegroups.com>
Mime-Version: 1.0
Content-Type: text/plain; charset=iso-8859-1
Content-Transfer-Encoding: 8bit
Injection-Info: flea.killfile.org; posting-host="port-212-4-179-246.dynamic.as20676.net:212.4.179.246";
logging-data="276067"; mail-complaints-to="news@news.killfile.org"
X-Mailer: Mozilla 4.75 [de] (Win98; U)
X-Accept-Language: de

View all headers

Sam Blake schrieb:
>
> On Saturday, January 13, 2024 at 1:56:38â¯PM UTC+11, Sam Blake wrote:
> >
> > ... and here's an example where Rubi outperforms both Mathematica
> > and my implementation of Mack's linear Hermite reduction (as given
> > in Symbolic Integration I by Manuel Bronstein)
> >
> > In[1151]:= (2 - 9 x^4 + 2 Sqrt[2] x^4 - 12 x^6 - 3 x^8)/(Sqrt[2] - 3 x^2 - x^4)^3;
> > HermiteReduce[%, x]
> > Integrate[%%, x]
> > Int[%%%, x]
> >
> > Out[1152]= {-((37549921350591017222596 (102743992 x +
> > 72655641 Sqrt[2] x))/((-8 + 9 Sqrt[2]) (-113 +
> > 72 Sqrt[2])^4 (2200 + 1593 Sqrt[2])^2 (-32544 +
> > 23137 Sqrt[2]) (3352883803641 +
> > 2370846461804 Sqrt[2]) (-3186 - 2200 Sqrt[2] + 6600 x^2 +
> > 4779 Sqrt[2] x^2 + 2200 x^4 + 1593 Sqrt[2] x^4))), 0}
> >
> > Out[1153]= -(1/(
> > 105 (-2 + 3 Sqrt[2] x^2 + Sqrt[2] x^4)^3))(Sqrt[2] - 3 x^2 -
> > x^4)^3 (210 x AppellF1[1/2, 3, 3, 3/2, (
> > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> > 3 + Sqrt[9 + 4 Sqrt[2]]))] +
> > 21 (-9 + 2 Sqrt[2]) x^5 AppellF1[5/2, 3, 3, 7/2, (
> > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> > 3 + Sqrt[9 + 4 Sqrt[2]]))] -
> > 5 (36 x^7 AppellF1[7/2, 3, 3, 9/2, (
> > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> > 3 + Sqrt[9 + 4 Sqrt[2]]))] +
> > 7 x^9 AppellF1[9/2, 3, 3, 11/2, (
> > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> > 3 + Sqrt[9 + 4 Sqrt[2]]))]))
> >
> > Out[1154]= x/(Sqrt[2] - 3 x^2 - x^4)
>
> For fairness, Rubi makes a mess of this one
>
> In[1578]:= Int[(-2 - 2*Sqrt[3]*x - 3*x^2 + 4*Sqrt[3]*x^3 + 7*x^4 + 6*Sqrt[3]*x^5 + 10*x^6 + 2*Sqrt[3]*x^7 + 6*x^8 + x^10)/(1 + 3*x^2 + Sqrt[3]*x^3 + 3*x^4 + x^6)^2, x]
>
> Out[1578]= -((3*3^(1/6) + 4*Sqrt[3] -
> 5*3^(5/6) + (3 - 4*3^(2/3))*
> x)/(3*(3 - 4*3^(2/3))*(1 + 3^(1/6)*x + x^2))) -
> (3^(1/3)*(6*I - 3*(-1)^(2/3)*3^(1/6) + 2*Sqrt[3] +
> 5*3^(5/6) + (-1)^(1/3)*(3 + 3^(2/3)*(2 + 2*I*Sqrt[3]))*x))/
> ((1 + (-1)^(1/3))^4*(4 - (-1)^(2/3)*3^(1/3))*(1 - (-1)^(1/
> 3)*3^(1/6)*x + x^2)) -
> (2*(-1)^(1/3)*(3*(-1)^(2/3)*3^(1/6) + 4*Sqrt[3] +
> 5*(-1)^(1/3)*3^(5/6) + (3 - 4*(-3)^(2/3))*x))/
> (3*3^(2/3)*(8 + 3^(1/3) + I*3^(5/6))*(1 + (-1)^(2/3)*3^(1/6)*x +
> x^2)) +
> (2*ArcTan[((-1)^(1/3)*3^(1/6) - 2*x)/
> Sqrt[4 - (-1)^(2/3)*3^(1/3)]])/(3*
> Sqrt[(1/2)*(8 + 3^(1/3) - I*3^(5/6))]) -
> (2*ArcTan[((-1)^(2/3)*3^(1/6) + 2*x)/Sqrt[4 + (-3)^(1/3)]])/(3*
> Sqrt[(1/2)*(8 + 3^(1/3) + I*3^(5/6))]) +
> (2*I*ArcTanh[(3^(1/6)*(I - Sqrt[3]) - 4*I*x)/
> Sqrt[2*(8 + 3^(1/3) - I*3^(5/6))]])/
> (3*Sqrt[(1/2)*(8 + 3^(1/3) - I*3^(5/6))]) +
> (2*I*ArcTanh[(3^(1/6)*(I + Sqrt[3]) - 4*I*x)/
> Sqrt[2*(8 + 3^(1/3) + I*3^(5/6))]])/
> (3*Sqrt[(1/2)*(8 + 3^(1/3) + I*3^(5/6))])
>

Derive 6.10 handles this in one step by one rule:

" INT(F'(x)/F(x)^n,x) -> -1/((n-1)*F(x)^(n-1)) "

Perhaps Rubi should be taught this rule as well?

Martin.

Subject: Re: An interesting rational integral in Mathematica
From: Sam Blake
Newsgroups: sci.math.symbolic
Date: Sat, 13 Jan 2024 09:51 UTC
References: 1 2 3 4

X-Received: by 2002:a05:6214:224:b0:681:554a:a94c with SMTP id j4-20020a056214022400b00681554aa94cmr10228qvt.3.1705139461096;
Sat, 13 Jan 2024 01:51:01 -0800 (PST)
X-Received: by 2002:a05:620a:278c:b0:783:54fe:5a10 with SMTP id
g12-20020a05620a278c00b0078354fe5a10mr3803qkp.5.1705139460813; Sat, 13 Jan
2024 01:51:00 -0800 (PST)
Path: eternal-september.org!news.eternal-september.org!feeder3.eternal-september.org!usenet.blueworldhosting.com!diablo1.usenet.blueworldhosting.com!peer02.iad!feed-me.highwinds-media.com!news.highwinds-media.com!news-out.google.com!nntp.google.com!postnews.google.com!google-groups.googlegroups.com!not-for-mail
Newsgroups: sci.math.symbolic
Date: Sat, 13 Jan 2024 01:51:00 -0800 (PST)
In-Reply-To: <65A257FB.690DA194@nowhere.invalid>
Injection-Info: google-groups.googlegroups.com; posting-host=144.130.168.222; posting-account=wlWA4gkAAABfPzIzZZgzF-hX5pK428zY
NNTP-Posting-Host: 144.130.168.222
References: <b3d304f9-640f-45b2-9cf6-24e7e2853753n@googlegroups.com>
<70c96453-8dd1-420a-b03c-3be0718b1182n@googlegroups.com> <c4a7b82c-90af-43f3-82bc-f2414c5583a0n@googlegroups.com>
<65A257FB.690DA194@nowhere.invalid>
User-Agent: G2/1.0
MIME-Version: 1.0
Message-ID: <3c69ac18-8120-48ca-bf21-f53dad39f7ebn@googlegroups.com>
Subject: Re: An interesting rational integral in Mathematica
From: samuel.thomas.blake@gmail.com (Sam Blake)
Injection-Date: Sat, 13 Jan 2024 09:51:01 +0000
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
X-Received-Bytes: 5765

View all headers

On Saturday, January 13, 2024 at 8:29:08 PM UTC+11, nob...@nowhere.invalid wrote:
> Sam Blake schrieb:
> >
> > On Saturday, January 13, 2024 at 1:56:38 PM UTC+11, Sam Blake wrote:
> > >
> > > ... and here's an example where Rubi outperforms both Mathematica
> > > and my implementation of Mack's linear Hermite reduction (as given
> > > in Symbolic Integration I by Manuel Bronstein)
> > >
> > > In[1151]:= (2 - 9 x^4 + 2 Sqrt[2] x^4 - 12 x^6 - 3 x^8)/(Sqrt[2] - 3 x^2 - x^4)^3;
> > > HermiteReduce[%, x]
> > > Integrate[%%, x]
> > > Int[%%%, x]
> > >
> > > Out[1152]= {-((37549921350591017222596 (102743992 x +
> > > 72655641 Sqrt[2] x))/((-8 + 9 Sqrt[2]) (-113 +
> > > 72 Sqrt[2])^4 (2200 + 1593 Sqrt[2])^2 (-32544 +
> > > 23137 Sqrt[2]) (3352883803641 +
> > > 2370846461804 Sqrt[2]) (-3186 - 2200 Sqrt[2] + 6600 x^2 +
> > > 4779 Sqrt[2] x^2 + 2200 x^4 + 1593 Sqrt[2] x^4))), 0}
> > >
> > > Out[1153]= -(1/(
> > > 105 (-2 + 3 Sqrt[2] x^2 + Sqrt[2] x^4)^3))(Sqrt[2] - 3 x^2 -
> > > x^4)^3 (210 x AppellF1[1/2, 3, 3, 3/2, (
> > > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> > > 3 + Sqrt[9 + 4 Sqrt[2]]))] +
> > > 21 (-9 + 2 Sqrt[2]) x^5 AppellF1[5/2, 3, 3, 7/2, (
> > > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> > > 3 + Sqrt[9 + 4 Sqrt[2]]))] -
> > > 5 (36 x^7 AppellF1[7/2, 3, 3, 9/2, (
> > > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> > > 3 + Sqrt[9 + 4 Sqrt[2]]))] +
> > > 7 x^9 AppellF1[9/2, 3, 3, 11/2, (
> > > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(
> > > 3 + Sqrt[9 + 4 Sqrt[2]]))]))
> > >
> > > Out[1154]= x/(Sqrt[2] - 3 x^2 - x^4)
> >
> > For fairness, Rubi makes a mess of this one
> >
> > In[1578]:= Int[(-2 - 2*Sqrt[3]*x - 3*x^2 + 4*Sqrt[3]*x^3 + 7*x^4 + 6*Sqrt[3]*x^5 + 10*x^6 + 2*Sqrt[3]*x^7 + 6*x^8 + x^10)/(1 + 3*x^2 + Sqrt[3]*x^3 + 3*x^4 + x^6)^2, x]
> >
> > Out[1578]= -((3*3^(1/6) + 4*Sqrt[3] -
> > 5*3^(5/6) + (3 - 4*3^(2/3))*
> > x)/(3*(3 - 4*3^(2/3))*(1 + 3^(1/6)*x + x^2))) -
> > (3^(1/3)*(6*I - 3*(-1)^(2/3)*3^(1/6) + 2*Sqrt[3] +
> > 5*3^(5/6) + (-1)^(1/3)*(3 + 3^(2/3)*(2 + 2*I*Sqrt[3]))*x))/
> > ((1 + (-1)^(1/3))^4*(4 - (-1)^(2/3)*3^(1/3))*(1 - (-1)^(1/
> > 3)*3^(1/6)*x + x^2)) -
> > (2*(-1)^(1/3)*(3*(-1)^(2/3)*3^(1/6) + 4*Sqrt[3] +
> > 5*(-1)^(1/3)*3^(5/6) + (3 - 4*(-3)^(2/3))*x))/
> > (3*3^(2/3)*(8 + 3^(1/3) + I*3^(5/6))*(1 + (-1)^(2/3)*3^(1/6)*x +
> > x^2)) +
> > (2*ArcTan[((-1)^(1/3)*3^(1/6) - 2*x)/
> > Sqrt[4 - (-1)^(2/3)*3^(1/3)]])/(3*
> > Sqrt[(1/2)*(8 + 3^(1/3) - I*3^(5/6))]) -
> > (2*ArcTan[((-1)^(2/3)*3^(1/6) + 2*x)/Sqrt[4 + (-3)^(1/3)]])/(3*
> > Sqrt[(1/2)*(8 + 3^(1/3) + I*3^(5/6))]) +
> > (2*I*ArcTanh[(3^(1/6)*(I - Sqrt[3]) - 4*I*x)/
> > Sqrt[2*(8 + 3^(1/3) - I*3^(5/6))]])/
> > (3*Sqrt[(1/2)*(8 + 3^(1/3) - I*3^(5/6))]) +
> > (2*I*ArcTanh[(3^(1/6)*(I + Sqrt[3]) - 4*I*x)/
> > Sqrt[2*(8 + 3^(1/3) + I*3^(5/6))]])/
> > (3*Sqrt[(1/2)*(8 + 3^(1/3) + I*3^(5/6))])
> >
> Derive 6.10 handles this in one step by one rule:
>
> " INT(F'(x)/F(x)^n,x) -> -1/((n-1)*F(x)^(n-1)) "
>
> Perhaps Rubi should be taught this rule as well?
>
> Martin.

Here's a similar example where Hermite reduction should return the integral without any logarithmic terms:

In[1793]:= (x^3 (2 + x^2))/(-2 + Sqrt[3] + (-2 + Sqrt[3]) x^2 - x^4)^2;
Integrate[%, x]
D[%, x] - %% // Simplify

Out[1794]= (((-2 - I) + Sqrt[3] - 2 x^2) ((1 + 2 I) - I Sqrt[3] +
2 I x^2) (x^4 (-2 I + ((2 - I) + Sqrt[3]) x^2) +
2 (-52 +
30 Sqrt[3] + ((-78 - 7 I) + (45 + 4 I) Sqrt[
3]) x^2 + ((-40 - 7 I) + (23 + 4 I) Sqrt[
3]) x^4 + ((-7 - 2 I) + (4 + I) Sqrt[3]) x^6) Log[(
2 (-2 + Sqrt[3]) + ((-2 - I) + Sqrt[3]) x^2)/(
2 (-2 + Sqrt[3]) + ((-2 + I) + Sqrt[3]) x^2)] +
2 (-52 +
30 Sqrt[3] + ((-78 - 7 I) + (45 + 4 I) Sqrt[
3]) x^2 + ((-40 - 7 I) + (23 + 4 I) Sqrt[
3]) x^4 + ((-7 - 2 I) + (4 + I) Sqrt[3]) x^6) Log[(
2 (-2 + Sqrt[3]) + ((-2 + I) + Sqrt[3]) x^2)/(
2 (-2 + Sqrt[3]) + ((-2 - I) + Sqrt[3]) x^2)]))/(8 (2 (-2 +
Sqrt[3]) + ((-2 - I) + Sqrt[3]) x^2) (-2 + Sqrt[
3] + (-2 + Sqrt[3]) x^2 - x^4)^2)

Out[1795]= 0

Subject: Re: An interesting rational integral in Mathematica
From: Sam Blake
Newsgroups: sci.math.symbolic
Date: Sat, 13 Jan 2024 09:52 UTC
References: 1 2 3 4

X-Received: by 2002:a05:6214:2aad:b0:681:2fe7:3f0f with SMTP id js13-20020a0562142aad00b006812fe73f0fmr232452qvb.8.1705139524361;
Sat, 13 Jan 2024 01:52:04 -0800 (PST)
X-Received: by 2002:a05:6214:405:b0:680:b69b:1be3 with SMTP id
z5-20020a056214040500b00680b69b1be3mr247547qvx.9.1705139524141; Sat, 13 Jan
2024 01:52:04 -0800 (PST)
Path: eternal-september.org!news.eternal-september.org!feeder3.eternal-september.org!panix!usenet.blueworldhosting.com!diablo1.usenet.blueworldhosting.com!peer02.iad!feed-me.highwinds-media.com!news.highwinds-media.com!news-out.google.com!nntp.google.com!postnews.google.com!google-groups.googlegroups.com!not-for-mail
Newsgroups: sci.math.symbolic
Date: Sat, 13 Jan 2024 01:52:03 -0800 (PST)
In-Reply-To: <65A257FB.690DA194@nowhere.invalid>
Injection-Info: google-groups.googlegroups.com; posting-host=144.130.168.222; posting-account=wlWA4gkAAABfPzIzZZgzF-hX5pK428zY
NNTP-Posting-Host: 144.130.168.222
References: <b3d304f9-640f-45b2-9cf6-24e7e2853753n@googlegroups.com>
<70c96453-8dd1-420a-b03c-3be0718b1182n@googlegroups.com> <c4a7b82c-90af-43f3-82bc-f2414c5583a0n@googlegroups.com>
<65A257FB.690DA194@nowhere.invalid>
User-Agent: G2/1.0
MIME-Version: 1.0
Message-ID: <3dbb13cc-a9d9-45c6-9e67-84b13d904bfen@googlegroups.com>
Subject: Re: An interesting rational integral in Mathematica
From: samuel.thomas.blake@gmail.com (Sam Blake)
Injection-Date: Sat, 13 Jan 2024 09:52:04 +0000
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
X-Received-Bytes: 4772

View all headers

Rubi gets the general form

In[1796]:= Int[f'[x]/f[x]^n, x]

Out[1796]= f[x]^(1 - n)/(1 - n)

Sam

A is for Apple. -- Hester Pryne

sci / sci.math.symbolic / Re: An interesting rational integral in Mathematica

Subject	Author
Re: An interesting rational integral in Mathematica	Sam Blake
Re: An interesting rational integral in Mathematica	Sam Blake
Re: An interesting rational integral in Mathematica	clicliclic@freenet.de
Re: An interesting rational integral in Mathematica	Sam Blake
Re: An interesting rational integral in Mathematica	Sam Blake
Re: An interesting rational integral in Mathematica	clicliclic@freenet.de