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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: 960
Registered: 4/26/08
Re: An independent integration test suite
Posted: Apr 21, 2013 12:20 PM
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Axel Vogt schrieb:
>
> Ok, for this 86 given integrals (translated to Maple syntax) Maple 17 solves all,
> BUT ONE, Int(exp(1/2*x)/(exp(x)-1)^(1/2),x) = 2*ln((exp(x)-1)^(1/2)+exp(1/2*x)).
>
> That can be done manually:
>
> Int(exp(1/2*x)/(exp(x)-1)^(1/2),x);
> Change(%, exp(1/2*x) = y, y);
> value(%);
> subs(y = exp(1/2*x), %); # re-substitute
> combine(%); # re-write, giving the asserted
>
> For the remaining 85 solutions it can automatically prove that lhs = rhs after
> differentiation w.r.t. x and it fails for no. 50, 52, 53:
>
> #Int(x/(a^4-x^4)^(1/2),x)
> 1/2*arctan(x^2/(a^4-x^4)^(1/2)) =
> 1/2*arcsin(x^2/a^2),
>
> #Int(1/(x*(a^2-x^2)^(1/2)),x)
> -1/(a^2)^(1/2)*ln((2*a^2+2*(a^2)^(1/2)*(a^2-x^2)^(1/2))/x) =
> -1/a*arctanh((a^2-x^2)^(1/2)/a),
>
> #Int(1/(x*(a^2+x^2)^(1/2)),x)
> -1/(a^2)^(1/2)*ln((2*a^2+2*(a^2)^(1/2)*(a^2+x^2)^(1/2))/x) =
> -1/a*arctanh(a/(a^2+x^2)^(1/2))
>
> Assuming a<>0 resolves the last two ones.
>
> For the first one I think that is false for a= exp(I*Pi/4),
> as already said in a previous post (20.04.2013)


I have repaired the evaluation of INT(x/SQRT(a^4-x^4)), x), it now
applies to complex 'a' as well. I also saw that some other evaluations
(i.e. 3, 4, 5, 55, 57, 71) could be shortened by converting from LN to
ATANH, and have made the substitutions. The full set of corrected
integrals from chapter 1 of Timofeev's book is appended below.

I am still counting integrals involving subcases as a single integral,
because splitting them into separate integrals would destroy the
correspondence with Timofeev's numbering. In order to allow easy
reference, the number would then have to be included explicitely, e.g.
as

["30a", INT(SIN(x)^2, x) = 1/2*(x - SIN(x)*COS(x))]

and vectors would still have to be parsed. I am therefore undecided on
this issue.

The following table summarizes the performance data made available so
far. I am no longer counting integral 29 (requiring a non-default
setting of a simplification flag) among the successes of Derive. I am
assuming that results produced by FriCAS never consist of error messages
or involve unresolved integrals or non-elementary functions. And I am
counting INT(EXP(x/2)/SQRT(EXP(x)-1), x) as a failure for Maple. Yes.

ch. ex's Der. Fri. Mpl MMA etc. etc.
6.10 1.20 17 ...
----------------------------------------------
1 81 78 81 80 ??
2 90 - - - -
3 14 - - - -
4 132 - - - -
5 120 - - - -
6 26 - - - -
7 11 - - - -
8 59 - - - -
--------------------------------------------
100% 96.3% 100% 98.8% ????%

The Winner so far is .............. FriCAS!

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/(2*a)*ATANH(1/CSC(2*a*x))=-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*x/4)-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*(a^2/SQRT(a^4-x^4))*SQRT(1-(x/a)^4)*A~
SIN((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)*LN(SQRT(3)*SQRT(3*x^2-4*x+5~
)+3*x-2)

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*(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
Read Re: An independent integration test suite
Nasser Abbasi
4/20/13
Read Re: An independent integration test suite
Rouben Rostamian
4/20/13
Read Re: An independent integration test suite
clicliclic@freenet.de
4/20/13
Read Re: An independent integration test suite
Rouben Rostamian
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
Read Re: An independent integration test suite
Axel Vogt
4/21/13
Read Re: An independent integration test suite
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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clicliclic@freenet.de
4/25/13
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Waldek Hebisch
4/26/13
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clicliclic@freenet.de
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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Peter Pein
4/26/13
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clicliclic@freenet.de
4/26/13
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Peter Pein
4/26/13
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clicliclic@freenet.de
4/26/13
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Richard Fateman
4/27/13
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clicliclic@freenet.de
4/27/13
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Richard Fateman
6/30/13
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clicliclic@freenet.de
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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Axel Vogt
7/1/13
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Waldek Hebisch
7/2/13
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7/2/13
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7/2/13
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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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7/4/13
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7/4/13
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Nasser Abbasi
7/5/13
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7/9/13
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7/10/13
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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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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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9/15/13
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9/21/13
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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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9/25/13
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Albert D. Rich
9/25/13
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9/25/13
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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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10/5/13
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10/6/13
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10/10/13
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Albert D. Rich
10/10/13
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Nasser Abbasi
10/11/13
Read Re: The A. F. Timofeev symbolic integration test suite
clicliclic@freenet.de
11/6/13
Read Re: The A. F. Timofeev symbolic integration test suite
Albert D. Rich
11/6/13
Read Re: The A. F. Timofeev symbolic integration test suite
Nasser Abbasi
11/7/13
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did
11/7/13
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11/7/13
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11/7/13
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11/12/13
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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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Albert D. Rich
11/14/13
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11/14/13
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Albert D. Rich
11/15/13
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11/15/13
Read Re: The A. F. Timofeev symbolic integration test suite
Albert D. Rich
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
clicliclic@freenet.de
11/21/13
Read Re: The A. F. Timofeev symbolic integration test suite
Albert D. Rich
11/21/13
Read Re: The A. F. Timofeev symbolic integration test suite
clicliclic@freenet.de
11/21/13
Read Re: The A. F. Timofeev symbolic integration test suite
Nasser Abbasi
11/21/13
Read Re: The A. F. Timofeev symbolic integration test suite
Albert D. Rich
11/21/13
Read Re: The A. F. Timofeev symbolic integration test suite
Albert D. Rich
11/22/13
Read Re: The A. F. Timofeev symbolic integration test suite
clicliclic@freenet.de
11/14/13
Read Re: The A. F. Timofeev symbolic integration test suite
Albert D. Rich
11/15/13
Read Re: The A. F. Timofeev symbolic integration test suite
clicliclic@freenet.de
11/15/13
Read Re: The A. F. Timofeev symbolic integration test suite
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
Read Re: An independent integration test suite
clicliclic@freenet.de
4/20/13
Read Re: An independent integration test suite
Axel Vogt
4/20/13
Read Re: An independent integration test suite
clicliclic@freenet.de
4/20/13
Read Re: An independent integration test suite
Waldek Hebisch
4/21/13
Read Re: An independent integration test suite
G. A. Edgar
12/8/13
Read Re: An independent integration test suite
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
clicliclic@freenet.de

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