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Topic: fyi, updated a CAS simplification comparing Mathematica with Maple
Replies: 13   Last Post: May 17, 2012 7:19 AM

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clicliclic@freenet.de

Posts: 974
Registered: 4/26/08
Re: fyi, updated a CAS simplification comparing Mathematica with Maple
Posted: May 17, 2012 2:03 AM
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Waldek Hebisch schrieb:
>
> clicliclic@freenet.de wrote:

> >
> > [...]
> >
> > However, even with SUBST( ,LN(SQRT(2)-1),-LN(SQRT(2)+1)), one does
> > not get the answer 1 easily. This is part of a general weakness of
> > Derive on quotients of algebraic expressions: one has to
> > "rationalize" the denominator by hand. (The first version of Rubi
> > had the same weakness, by the way).
> >

>
> Hmm, by default FriCAS keeps algebraics in denominator. However
> when quotient is rational then "rational" GCD is enough to get 1.
> OTOH when quotient is irrational by defaults one gets unsimplified
> quotient (containing the last remaining log). In such case one
> has to turn on a flag to trigger removing irrationalities from
> denominators (and only after that the last log vanishes).
>


In Derive 6.10 practice, the simplification looks as follows: Putting
the expanded (xnum/xden)^32 on a common denominator, one has:

(1572584048032918633353217-1111984844349868137938112*SQRT(2))*(L~
N(SQRT(2)-1)^32-32*SQRT(2)*LN(SQRT(2)-1)^31+992*LN(SQRT(2)-1)^30~
-9920*SQRT(2)*LN(SQRT(2)-1)^29+143840*LN(SQRT(2)-1)^28-805504*SQ~
RT(2)*LN(SQRT(2)-1)^27+7249536*LN(SQRT(2)-1)^26-26926848*SQRT(2)~
*LN(SQRT(2)-1)^25+168292800*LN(SQRT(2)-1)^24-448780800*SQRT(2)*L~
N(SQRT(2)-1)^23+2064391680*LN(SQRT(2)-1)^22-4128783360*SQRT(2)*L~
N(SQRT(2)-1)^21+14450741760*LN(SQRT(2)-1)^20-22231910400*SQRT(2)~
*LN(SQRT(2)-1)^19+60343756800*LN(SQRT(2)-1)^18-72412508160*SQRT(~
2)*LN(SQRT(2)-1)^17+153876579840*LN(SQRT(2)-1)^16-144825016320*S~
QRT(2)*LN(SQRT(2)-1)^15+241375027200*LN(SQRT(2)-1)^14-1778552832~
00*SQRT(2)*LN(SQRT(2)-1)^13+231211868160*LN(SQRT(2)-1)^12-132121~
067520*SQRT(2)*LN(SQRT(2)-1)^11+132121067520*LN(SQRT(2)-1)^10-57~
443942400*SQRT(2)*LN(SQRT(2)-1)^9+43082956800*LN(SQRT(2)-1)^8-13~
786546176*SQRT(2)*LN(SQRT(2)-1)^7+7423524864*LN(SQRT(2)-1)^6-164~
9672192*SQRT(2)*LN(SQRT(2)-1)^5+589168640*LN(SQRT(2)-1)^4-812646~
40*SQRT(2)*LN(SQRT(2)-1)^3+16252928*LN(SQRT(2)-1)^2-1048576*SQRT~
(2)*LN(SQRT(2)-1)+65536)/((2*SQRT(2)-3)*LN(SQRT(2)+1)-3*SQRT(2)+~
4)^32

The usual Derive commands and flags are of no help in collapsing this.
Applying SUBST( ,LN(SQRT(2)-1),-LN(SQRT(2)+1)) one has:

(1572584048032918633353217-1111984844349868137938112*SQRT(2))*(L~
N(SQRT(2)+1)^32+32*SQRT(2)*LN(SQRT(2)+1)^31+992*LN(SQRT(2)+1)^30~
+9920*SQRT(2)*LN(SQRT(2)+1)^29+143840*LN(SQRT(2)+1)^28+805504*SQ~
RT(2)*LN(SQRT(2)+1)^27+7249536*LN(SQRT(2)+1)^26+26926848*SQRT(2)~
*LN(SQRT(2)+1)^25+168292800*LN(SQRT(2)+1)^24+448780800*SQRT(2)*L~
N(SQRT(2)+1)^23+2064391680*LN(SQRT(2)+1)^22+4128783360*SQRT(2)*L~
N(SQRT(2)+1)^21+14450741760*LN(SQRT(2)+1)^20+22231910400*SQRT(2)~
*LN(SQRT(2)+1)^19+60343756800*LN(SQRT(2)+1)^18+72412508160*SQRT(~
2)*LN(SQRT(2)+1)^17+153876579840*LN(SQRT(2)+1)^16+144825016320*S~
QRT(2)*LN(SQRT(2)+1)^15+241375027200*LN(SQRT(2)+1)^14+1778552832~
00*SQRT(2)*LN(SQRT(2)+1)^13+231211868160*LN(SQRT(2)+1)^12+132121~
067520*SQRT(2)*LN(SQRT(2)+1)^11+132121067520*LN(SQRT(2)+1)^10+57~
443942400*SQRT(2)*LN(SQRT(2)+1)^9+43082956800*LN(SQRT(2)+1)^8+13~
786546176*SQRT(2)*LN(SQRT(2)+1)^7+7423524864*LN(SQRT(2)+1)^6+164~
9672192*SQRT(2)*LN(SQRT(2)+1)^5+589168640*LN(SQRT(2)+1)^4+812646~
40*SQRT(2)*LN(SQRT(2)+1)^3+16252928*LN(SQRT(2)+1)^2+1048576*SQRT~
(2)*LN(SQRT(2)+1)+65536)/((2*SQRT(2)-3)*LN(SQRT(2)+1)-3*SQRT(2)+~
4)^32

But the usual commands and flags are again of no help. However, with
((2*SQRT(2)-3)*LN(SQRT(2)+1)+3*SQRT(2)-4)^32 inserted in both numerator
and denominator, this expression collapses automatically:

(1572584048032918633353217-1111984844349868137938112*SQRT(2))*(L~
N(SQRT(2)+1)^32+32*SQRT(2)*LN(SQRT(2)+1)^31+992*LN(SQRT(2)+1)^30~
+9920*SQRT(2)*LN(SQRT(2)+1)^29+143840*LN(SQRT(2)+1)^28+805504*SQ~
RT(2)*LN(SQRT(2)+1)^27+7249536*LN(SQRT(2)+1)^26+26926848*SQRT(2)~
*LN(SQRT(2)+1)^25+168292800*LN(SQRT(2)+1)^24+448780800*SQRT(2)*L~
N(SQRT(2)+1)^23+2064391680*LN(SQRT(2)+1)^22+4128783360*SQRT(2)*L~
N(SQRT(2)+1)^21+14450741760*LN(SQRT(2)+1)^20+22231910400*SQRT(2)~
*LN(SQRT(2)+1)^19+60343756800*LN(SQRT(2)+1)^18+72412508160*SQRT(~
2)*LN(SQRT(2)+1)^17+153876579840*LN(SQRT(2)+1)^16+144825016320*S~
QRT(2)*LN(SQRT(2)+1)^15+241375027200*LN(SQRT(2)+1)^14+1778552832~
00*SQRT(2)*LN(SQRT(2)+1)^13+231211868160*LN(SQRT(2)+1)^12+132121~
067520*SQRT(2)*LN(SQRT(2)+1)^11+132121067520*LN(SQRT(2)+1)^10+57~
443942400*SQRT(2)*LN(SQRT(2)+1)^9+43082956800*LN(SQRT(2)+1)^8+13~
786546176*SQRT(2)*LN(SQRT(2)+1)^7+7423524864*LN(SQRT(2)+1)^6+164~
9672192*SQRT(2)*LN(SQRT(2)+1)^5+589168640*LN(SQRT(2)+1)^4+812646~
40*SQRT(2)*LN(SQRT(2)+1)^3+16252928*LN(SQRT(2)+1)^2+1048576*SQRT~
(2)*LN(SQRT(2)+1)+65536)*((2*SQRT(2)-3)*LN(SQRT(2)+1)+3*SQRT(2)-~
4)^32/(((2*SQRT(2)-3)*LN(SQRT(2)+1)-3*SQRT(2)+4)^32*((2*SQRT(2)-~
3)*LN(SQRT(2)+1)+3*SQRT(2)-4)^32)

1

Martin.



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