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Topic: An independent integration test suite
Replies: 128   Last Post: Dec 8, 2013 3:21 PM

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: An independent integration test suite / Chap 3
Posted: Jul 1, 2013 2:12 PM

Axel Vogt schrieb:
>
> On 30.06.2013 18:07, clicliclic@freenet.de wrote:

> >
> > The following table summarizes the performance data made available for
> > the example integrals in Timofeev's 1948 book:
> >
> >
> > To simplify the accounting, the numbers of examples represent the actual
> > numbers of distinct integrands; for chapters where this has not yet been
> > determined, the total of Timofeev's numbered examples is given instead:
> >
> > ch. ex's Der. Fri. Mpl MMA etc. etc.
> > 6.10 1.20 17 9.xx
> > -----------------------------------------------
> > 1 86 83 85 85 86
> > 2 90 - - - -
> > 3 12 12 ? ? ?
> > 4 132 - - - -
> > 5 120 - - - -
> > 6 26 - - - -
> > 7 11 9 ? 11 ?
> > 8 109 - - - -
> > 9 59 - - - -
> > -----------------------------------------------
> > 100% 95.4% ? % ? % ? %
> >

> ...
>
> Would you mind to give the Derive input code for Chapter 3 again?
> Sorry, I am a bit lost in the lengthy thread.

Just in case, below are the Derive versions of the example integrals
from both chapter 3 and chapter 7.

Martin.

" Timofeev (1948) Ch. 3, examples 1 - 3 (p. 101) ... "

INT(1/((x-2)^3*(x+1)^2),x)=(2*x^2-5*x-1)/(18*(x+1)*(x-2)^2)+1/27~
*LN((x-2)/(x+1))

INT(1/((x+2)^3*(x+3)^4),x)=(60*x^4+630*x^3+2450*x^2+4175*x+2627)~
/(6*(x+2)^2*(x+3)^3)+10*LN((x+2)/(x+3))

INT(x^5/(3+x)^2,x)=1/4*x^4-2*x^3+27/2*x^2-108*x+243/(x+3)+405*LN~
(x+3)

" Timofeev (1948) Ch. 3, examples 4 - 9 (p. 105) ... "

[]

[]

INT(x/(3+6*x+2*x^2),x)=1/4*LN(-(3+6*x+2*x^2))+SQRT(3)/2*ATANH((3~
+2*x)/SQRT(3))=(1/4-SQRT(3)/4)*LN(2*x-SQRT(3)+3)+(SQRT(3)/4+1/4)~
*LN(2*x+SQRT(3)+3)

INT((2*x-3)/(3+6*x+2*x^2)^3,x)=-(8*x^3+36*x^2+44*x+13)/(4*(2*x^2~
+6*x+3)^2)+1/SQRT(3)*ATANH((3+2*x)/SQRT(3))

INT((x-1)/(x^2+5*x+4)^2,x)=(7*x+13)/(9*(x^2+5*x+4))+7/27*LN((x+1~
)/(x+4))

INT(1/(x^2+3*x+2)^5,x)=(2*x+3)/(4*(x^2+3*x+2)^4)*(-1+14/3*(x^2+3~
*x+2)-70/3*(x^2+3*x+2)^2+140*(x^2+3*x+2)^3)+70*LN((x+1)/(x+2))

" Timofeev (1948) Ch. 3, examples 10 - 12 (p. 109) ... "

INT(1/(x^3*(7-6*x+2*x^2)^2),x)=-1/(98*x^2)-12/(343*x)+2*(41-9*x)~
/(1715*(7-6*x+2*x^2))+80/2401*LN(x)-40/2401*LN(7-6*x+2*x^2)+234*~
SQRT(5)/60025*ATAN((2*x-3)/SQRT(5))

INT(x^9/(x^2+3*x+2)^5,x)=-(25*x^8+35292*x^7+369950*x^6+1632276*x~
^5+3919731*x^4+5527800*x^3+4578216*x^2+2063520*x+390960)/(24*(x^~
2+3*x+2)^4)+1472*LN(x+2)-1471*LN(x+1)

INT((1+2*x)^2/(3+5*x+2*x^2)^5,x)=-(11+10*x)/(4*(2*x^2+5*x+3)^4)+~
31*(5+4*x)/(6*(2*x^2+5*x+3)^3)*(1-10*(2*x^2+5*x+3)+120*(2*x^2+5*~
x+3)^2)+2480*LN((x+1)/(2*x+3))

" Timofeev (1948) Ch. 3, examples 13 - 14 (p. 113) ... "

INT((a-b*x^2)^3/x^7,x)=-a^3/(6*x^6)+3*a^2*b/(4*x^4)-3*a*b^2/(2*x~
^2)-b^3*LN(x)

INT(x^13/(a^4+x^4)^5,x)=x^2*(15*x^12-73*a^4*x^8-55*a^8*x^4-15*a^~
12)/(768*a^4*(x^4+a^4)^4)+5/(256*a^6)*ATAN(x^2/a^2)

" ... end of Timofeev Ch. 3 "

" Timofeev (1948) Ch. 7, examples 1 - 4 (p. 334) ... "

INT(x^2*COS(x)^5,x)=1/200*x*COS(5*x)+(1/80*x^2-1/1000)*SIN(5*x)+~
5/72*x*COS(3*x)+(5/48*x^2-5/216)*SIN(3*x)+5/4*x*COS(x)+(5/8*x^2-~
5/4)*SIN(x)

INT(x^3*SIN(x)^3,x)=1/12*(x^3-2/3*x)*COS(3*x)-1/12*(x^2-2/9)*SIN~
(3*x)-3/4*(x^3-6*x)*COS(x)+9/4*(x^2-2)*SIN(x)

INT(x^2*SIN(x)^6,x)=5/48*x^3-1/192*(x^2-1/18)*SIN(6*x)-1/576*x*C~
OS(6*x)+3/64*(x^2-1/8)*SIN(4*x)+3/128*x*COS(4*x)-15/64*(x^2-1/2)~
*SIN(2*x)-15/64*x*COS(2*x)

INT(x^2*SIN(x)^2*COS(x),x)=1/3*x^2*SIN(x)^3-1/18*x*COS(3*x)+1/54~
*SIN(3*x)+1/2*x*COS(x)-1/2*SIN(x)

" Timofeev (1948) Ch. 7, examples 5 - 9 (p. 342-343) ... "

INT(x*COS(x)^4/SIN(x)^2,x)=-x*COS(x)*(1/2*SIN(x)+1/SIN(x))+1/4*S~
IN(x)^2+LN(SIN(x))-3/4*x^2

INT(x*SIN(x)^3/COS(x)^4,x)=x*(1/(3*COS(x)^3)-1/COS(x))-SIN(x)/(6~
*COS(x)^2)+5/6*ATANH(SIN(x))

INT(x*SIN(x)/COS(x)^3,x)=x/(2*COS(x)^2)-1/2*TAN(x)

INT(x*SIN(x)^3/COS(x),x)=1/4*x*COS(2*x)-1/8*SIN(2*x)+INT(x*TAN(x~
),x)

INT(x*SIN(x)^3/COS(x)^3,x)=x/(2*COS(x)^2)-1/2*TAN(x)-INT(x*TAN(x~
),x)

" Timofeev (1948) Ch. 7, examples 10 - 11 (p. 344) ... "

INT((2*x+SIN(2*x))/(x*SIN(x)+COS(x))^2,x)=-2*COS(x)/(x*SIN(x)+CO~
S(x))

INT((x/(x*COS(x)-SIN(x)))^2,x)=(x*SIN(x)+COS(x))/(x*COS(x)-SIN(x~
))

" ... end of Timofeev Ch. 7 "

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