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

 Messages: [ Previous | Next ]
 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: An independent integration test suite
Posted: Jul 5, 2013 12:42 PM

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 clicliclic@freenet.de
3/19/13 clicliclic@freenet.de
3/21/13 Waldek Hebisch
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7/4/13 Nasser Abbasi
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7/10/13 Nasser Abbasi
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7/10/13 clicliclic@freenet.de
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9/21/13 Albert D. Rich
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9/25/13 Albert D. Rich
9/25/13 Albert D. Rich
9/25/13 clicliclic@freenet.de
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9/26/13 Albert D. Rich
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10/1/13 Albert D. Rich
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10/6/13 clicliclic@freenet.de
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10/10/13 Nasser Abbasi
10/11/13 clicliclic@freenet.de
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11/6/13 Nasser Abbasi
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11/14/13 Albert D. Rich
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11/15/13 Albert D. Rich
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11/16/13 clicliclic@freenet.de
11/21/13 Albert D. Rich
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11/21/13 Nasser Abbasi
11/21/13 Albert D. Rich
11/21/13 Albert D. Rich
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4/20/13 Richard Fateman
4/21/13 clicliclic@freenet.de
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4/20/13 Waldek Hebisch
4/21/13 G. A. Edgar
12/8/13 clicliclic@freenet.de
10/5/13 Albert D. Rich
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