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Topic: An independent integration test suite
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clicliclic@freenet.de

Posts: 988
Registered: 4/26/08
Re: An independent integration test suite
Posted: Jul 5, 2013 12:42 PM
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Thanks for the Maple, FriCAS, Mathematica, and Rubi performance data.

The following table summarizes the available data 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 Rubi etc. etc.
6.10 1.21 17 9.xx 4.1
-------------------------------------------------------
1 86 83 86 85 86 ?
2 90 - - - - -
3 12 12 12 12 12 12
4 132 - - - - -
5 120 - - - - -
6 26 - - - - -
7 11 9 11 11 11 8
8 109 - - - - -
9 59 - - - - -
-------------------------------------------------------
100% 95.4% 100% 99.1% 100% ?

The Winners so far are FriCAS 1.21 and Mathematica 9!

Only Rubi's performance on the problems from Chapters 1 remains unknown
now. Derive versions of these problems are appended below (taken from my
current file).

Martin.


" Timofeev (1948) Ch. 1, examples 1 - 21 (p. 25-26) ... "

INT(1/(a^2-b^2*x^2),x)=1/(a*b)*ATANH(b*x/a)

INT(1/(a^2+b^2*x^2),x)=1/(a*b)*ATAN(b*x/a)

INT(SEC(2*a*x),x)=1/(4*a)*LN((CSC(2*a*x)+1)/(CSC(2*a*x)-1))=1/(2~
*a)*LN(TAN(pi/4+a*x))

INT(1/(4*SIN(x/3)),x)=-3/4*ATANH(COS(x/3))=3/4*LN(TAN(x/6))

INT(1/COS(3/4*pi-2*x),x)=-1/2*ATANH(SIN(3/4*pi-2*x))=1/2*LN(TAN(~
pi/8-x))

INT(SEC(x)*TAN(x),x)=SEC(x)

INT(CSC(x)*COT(x),x)=-CSC(x)

INT(TAN(x)/SIN(2*x),x)=1/2*TAN(x)

INT(1/(1+COS(x)),x)=SIN(x)/(1+COS(x))=TAN(x/2)

INT(1/(1-COS(x)),x)=SIN(x)/(COS(x)-1)=-COT(x/2)

INT(SIN(x)/(a-b*COS(x)),x)=1/b*LN(a-b*COS(x))

INT(COS(x)/(a^2+b^2*SIN(x)^2),x)=1/(a*b)*ATAN(b*SIN(x)/a)

INT(COS(x)/(a^2-b^2*SIN(x)^2),x)=1/(a*b)*ATANH(b*SIN(x)/a)

[INT(SIN(2*x)/(b^2*SIN(x)^2+a^2),x)=1/b^2*LN(a^2+b^2*SIN(x)^2),I~
NT(SIN(2*x)/(b^2*SIN(x)^2-a^2),x)=1/b^2*LN(a^2-b^2*SIN(x)^2)]

[INT(SIN(2*x)/(b^2*COS(x)^2+a^2),x)=-1/b^2*LN(a^2+b^2*COS(x)^2),~
INT(SIN(2*x)/(b^2*COS(x)^2-a^2),x)=-1/b^2*LN(a^2-b^2*COS(x)^2)]

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

INT(#e^x/(#e^(2*x)-1),x)=-ATANH(#e^x)

INT(1/(x*LN(x)),x)=LN(LN(x))

INT(1/(x*(1+LN(x)^2)),x)=ATAN(LN(x))

INT(1/(x*(1-LN(x))),x)=-LN(1-LN(x))

INT(1/(x*(1+LN(x/a))),x)=LN(1+LN(x/a))

" Timofeev (1948) Ch. 1, examples 22 - 40 (p. 27-28) ... "

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

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

INT((2*x-1)/(2*x+3),x)=x-2*LN(2*x+3)

INT((2*x-5)/(3*x^2-2),x)=1/3*LN(2-3*x^2)+5/SQRT(6)*ATANH(SQRT(6)~
*x/2)=(4-5*SQRT(6))/12*LN(SQRT(3)*x-SQRT(2))+(5*SQRT(6)+4)/12*LN~
(SQRT(3)*x+SQRT(2))

INT((2*x-5)/(3*x^2+2),x)=1/3*LN(3*x^2+2)-5/SQRT(6)*ATAN(SQRT(6)*~
x/2)

INT(SIN(x)*SIN(x/4),x)=2/3*SIN(3*x/4)-2/5*SIN(5*x/4)

INT(COS(3*x)*COS(4*x),x)=1/14*SIN(7*x)+1/2*SIN(x)

INT(TAN(x)*TAN(x-a),x)=1/TAN(a)*LN(1+TAN(a)*TAN(x))-x

[INT(SIN(x)^2,x)=1/2*(x-SIN(x)*COS(x)),INT(COS(x)^2,x)=1/2*(x+SI~
N(x)*COS(x))]

INT(SIN(x)*COS(x)^3,x)=-1/4*COS(x)^4

INT(COS(x)^3/SIN(x)^4,x)=1/SIN(x)-1/(3*SIN(x)^3)

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

INT(COT(3/4*x)^2,x)=-4/3*COT(3/4*x)-x

INT((1+TAN(2*x))^2,x)=1/2*TAN(2*x)-LN(COS(2*x))

INT((TAN(x)-COT(x))^2,x)=TAN(x)-COT(x)-4*x

INT((TAN(x)-SEC(x))^2,x)=2*(TAN(x)-SEC(x))-x=2*TAN(x/2-pi/4)-x

INT(SIN(x)/(1+SIN(x)),x)=COS(x)/(1+SIN(x))+x=TAN(pi/4-x/2)+x

INT(COS(x)/(1-COS(x)),x)=SIN(x)/(COS(x)-1)-x=-COT(x/2)-x

INT((#e^(x/2)-1)^3*#e^(-x/2),x)=-6*#e^(x/2)+2*#e^(-x/2)+#e^x+3*x

" Timofeev (1948) Ch. 1, examples 41 - 65 (p. 35-37) ... "

INT(1/(x^2-6*x+5),x)=1/4*LN((x-5)/(x-1))

INT(x^2/(13-6*x^3+x^6),x)=1/6*ATAN((x^3-3)/2)

INT((x+2)/(x^2-4*x-1),x)=1/2*LN(1+4*x-x^2)+4/SQRT(5)*ATANH((2-x)~
/SQRT(5))=1/10*((4*SQRT(5)+5)*LN(x-SQRT(5)-2)+(5-4*SQRT(5))*LN(x~
+SQRT(5)-2))

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

INT(1/((a*x+b)*SQRT(x)),x)=2/(SQRT(a)*SQRT(b))*ATAN(SQRT(a)*SQRT~
(x)/SQRT(b))

INT(x^3*SQRT(1+x^2),x)=1/15*(3*x^4+x^2-2)*SQRT(x^2+1)

INT(x/SQRT(a^4-x^4),x)=1/2*ATAN(x^2/SQRT(a^4-x^4))=1/2*(a^2/SQRT~
(a^4-x^4))*SQRT(1-(x/a)^4)*ASIN((x/a)^2)

[INT(1/(x*SQRT(x^2-a^2)),x)=1/a*ATAN(SQRT(x^2-a^2)/a),INT(1/(x*S~
QRT(a^2-x^2)),x)=-1/a*ATANH(SQRT(a^2-x^2)/a),INT(1/(x*SQRT(x^2+a~
^2)),x)=-1/a*ATANH(a/SQRT(x^2+a^2))]

INT(1/SQRT(2+x-x^2),x)=ASIN((2*x-1)/3)

INT(1/SQRT(3*x^2-4*x+5),x)=1/SQRT(3)*ASINH((3*x-2)/SQRT(11))

INT(1/SQRT(x-x^2),x)=ASIN(2*x-1)

INT((2*x+1)/SQRT(2+x-x^2),x)=2*ASIN((2*x-1)/3)-2*SQRT(-x^2+x+2)

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

INT(1/((x-2)*SQRT(2+x-x^2)),x)=2*SQRT(2+x-x^2)/(3*(x-2))

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

INT(1/(2+3*COS(x)^2),x)=1/SQRT(10)*(x-ATAN(3*SIN(x)*COS(x)/(SQRT~
(10)+2+3*COS(x)^2)))

INT((1-TAN(x))/SIN(2*x),x)=-1/2*(ATANH(COS(2*x))+TAN(x))=1/2*(LN~
(TAN(x))-TAN(x))

INT((1+TAN(x)^2)/(1-TAN(x)^2),x)=1/2*LN((1+TAN(x))/(1-TAN(x)))

INT((a^2-4*COS(x)^2)^(3/4)*SIN(2*x),x)=1/7*(a^2-4*COS(x)^2)^(7/4)

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

INT(1/SQRT(a^(2*x)-1),x)=1/LN(a)*ATAN(SQRT(a^(2*x)-1))=1/LN(a)*A~
SEC(SQRT(a^(2*x)))

INT(#e^(x/2)/SQRT(#e^x-1),x)=2*LN(SQRT(#e^x-1)+#e^(x/2))

INT(ATAN(x)^n/(1+x^2),x)=1/(n+1)*ATAN(x)^(n+1)

INT(ASIN(x/a)^(3/2)/SQRT(a^2-x^2),x)=2/5*ASIN(x/a)^(3/2)*ATAN(x/~
SQRT(a^2-x^2))=2/5*(a/SQRT(a^2-x^2))*SQRT(1-(x/a)^2)*ASIN(x/a)^(~
5/2)

INT(1/(ACOS(x)^3*SQRT(1-x^2)),x)=1/(2*ACOS(x)^2)

" Timofeev (1948) Ch. 1, examples 66 - 81 (p. 41-42) ... "

INT(LN(x)^2*x,x)=x^2/2*(LN(x)^2-LN(x)+1/2)

INT(LN(x)/x^5,x)=-(4*LN(x)+1)/(16*x^4)

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

INT(COS(x)^5,x)=SIN(x)/15*(3*COS(x)^4+4*COS(x)^2+8)

INT(SIN(x)^2*COS(x)^4,x)=1/6*SIN(x)^3*COS(x)^3+1/8*SIN(x)^3*COS(~
x)-1/16*SIN(x)*COS(x)+x/16

INT(1/SIN(x)^5,x)=-3/8*ATANH(COS(x))-3*COS(x)/(8*SIN(x)^2)-COS(x~
)/(4*SIN(x)^4)

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

INT(#e^(2*x)*SIN(3*x),x)=1/13*#e^(2*x)*(2*SIN(3*x)-3*COS(3*x))

INT(a^x*COS(x),x)=a^x/(LN(a)^2+1)*(LN(a)*COS(x)+SIN(x))

INT(COS(LN(x)),x)=x/2*(COS(LN(x))+SIN(LN(x)))

INT(SEC(x)^2*LN(COS(x)),x)=TAN(x)*LN(COS(x))+TAN(x)-x

INT(x*TAN(x)^2,x)=LN(COS(x))+x*TAN(x)-1/2*x^2

INT(ASIN(x)/x^2,x)=-ASIN(x)/x+LN((1-SQRT(1-x^2))/x)

INT(ASIN(x)^2,x)=x*ASIN(x)^2+2*SQRT(1-x^2)*ASIN(x)-2*x

INT(x^2*ATAN(x)/(1+x^2),x)=x*ATAN(x)-1/2*ATAN(x)^2-1/2*LN(x^2+1)

INT(ACOS(SQRT(x/(x+1))),x)=(x+1)*(ACOS(SQRT(x/(x+1)))+SQRT(1/(x+~
1))*SQRT(x/(x+1)))

" ... end of Timofeev Ch. 1 "


Date Subject Author
2/24/13
Read An independent integration test suite
clicliclic@freenet.de
3/19/13
Read where the air gets thin for Axiom & Co.
clicliclic@freenet.de
3/21/13
Read Re: where the air gets thin for Axiom & Co.
Waldek Hebisch
3/22/13
Read Re: where the air gets thin for Axiom & Co.
clicliclic@freenet.de
3/26/13
Read Re: where the air gets thin for Axiom & Co.
Waldek Hebisch
3/26/13
Read Re: where the air gets thin for Axiom & Co.
clicliclic@freenet.de
4/20/13
Read Re: An independent integration test suite
clicliclic@freenet.de
4/20/13
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Nasser Abbasi
4/20/13
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Rouben Rostamian
4/20/13
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clicliclic@freenet.de
4/20/13
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Rouben Rostamian
4/20/13
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Axel Vogt
4/20/13
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clicliclic@freenet.de
4/20/13
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Axel Vogt
4/21/13
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Axel Vogt
4/21/13
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clicliclic@freenet.de
4/21/13
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Waldek Hebisch
4/22/13
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clicliclic@freenet.de
4/22/13
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Axel Vogt
4/22/13
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clicliclic@freenet.de
4/23/13
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Waldek Hebisch
4/24/13
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4/25/13
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Waldek Hebisch
4/26/13
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4/27/13
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Waldek Hebisch
4/24/13
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Richard Fateman
4/24/13
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clicliclic@freenet.de
4/25/13
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Richard Fateman
4/26/13
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clicliclic@freenet.de
4/26/13
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Axel Vogt
4/27/13
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clicliclic@freenet.de
4/25/13
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Waldek Hebisch
4/25/13
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Peter Pein
4/25/13
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Nasser Abbasi
4/26/13
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4/26/13
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clicliclic@freenet.de
4/26/13
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4/26/13
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clicliclic@freenet.de
4/26/13
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4/27/13
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4/27/13
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Richard Fateman
6/30/13
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6/30/13
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Axel Vogt
7/1/13
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clicliclic@freenet.de
7/1/13
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7/1/13
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Waldek Hebisch
7/2/13
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clicliclic@freenet.de
7/2/13
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7/2/13
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clicliclic@freenet.de
7/2/13
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Nasser Abbasi
7/2/13
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Nasser Abbasi
7/4/13
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clicliclic@freenet.de
7/4/13
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Nasser Abbasi
7/4/13
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Nasser Abbasi
7/5/13
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7/5/13
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Nasser Abbasi
7/9/13
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7/10/13
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Nasser Abbasi
7/10/13
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Richard Fateman
7/10/13
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Nasser Abbasi
7/10/13
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clicliclic@freenet.de
8/6/13
Read Rubi 4.1 and the Timofeev test suite
clicliclic@freenet.de
9/15/13
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Albert D. Rich
9/15/13
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clicliclic@freenet.de
9/15/13
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clicliclic@freenet.de
9/21/13
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Albert D. Rich
9/21/13
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9/22/13
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9/24/13
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9/30/13
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9/22/13
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Albert D. Rich
9/25/13
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Albert D. Rich
9/25/13
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Albert D. Rich
9/25/13
Read Re: Rubi 4.1 and the Timofeev test suite
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9/25/13
Read Re: Rubi 4.1 and the Timofeev test suite
Albert D. Rich
9/26/13
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Albert D. Rich
9/26/13
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9/26/13
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9/29/13
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10/1/13
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10/1/13
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10/1/13
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10/5/13
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10/6/13
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10/10/13
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10/10/13
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Nasser Abbasi
10/11/13
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11/6/13
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11/6/13
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11/7/13
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did
11/7/13
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11/7/13
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11/12/13
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11/13/13
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11/13/13
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11/14/13
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11/14/13
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11/15/13
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11/16/13
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11/16/13
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11/21/13
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11/21/13
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11/21/13
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Nasser Abbasi
11/21/13
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Albert D. Rich
11/21/13
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Albert D. Rich
11/22/13
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11/14/13
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Albert D. Rich
11/15/13
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clicliclic@freenet.de
11/15/13
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Nasser Abbasi
11/16/13
Read Re: The A. F. Timofeev symbolic integration test suite
clicliclic@freenet.de
11/16/13
Read Re: The A. F. Timofeev symbolic integration test suite
Nasser Abbasi
11/7/13
Read Re: The A. F. Timofeev symbolic integration test suite
did
11/7/13
Read Re: The A. F. Timofeev symbolic integration test suite
clicliclic@freenet.de
4/20/13
Read Re: An independent integration test suite
Richard Fateman
4/21/13
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clicliclic@freenet.de
4/20/13
Read Re: An independent integration test suite
Axel Vogt
4/20/13
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clicliclic@freenet.de
4/20/13
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Waldek Hebisch
4/21/13
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G. A. Edgar
12/8/13
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clicliclic@freenet.de
10/5/13
Read Re: An independent integration test suite
Albert D. Rich
10/6/13
Read Re: An independent integration test suite
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