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Topic: first anniversary of the IITS
Replies: 21   Last Post: Jan 10, 2015 4:01 PM

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 clicliclic@freenet.de Posts: 1,191 Registered: 4/26/08
first anniversary of the IITS
Posted: Feb 24, 2014 12:45 PM

To celebrate the 1st anniversary of the Independent Integration Test
Suite, I am offering streamlined versions of some evaluations given on
the Rubi website <http://www.apmaths.uwo.ca/~arich/> for the example
integrals from Chapter 5 of Timofeev's book.

It has been a disappointment to realize that the radicands appearing in
Timofeev's examples no. 85, 89-90, 98-99, and 109 from Chapter 5 are
negative over the entire real axis.

- Example 40 (p. 254) with logarithm arguments replicating the
denominator factors (1st version):

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

- Example 44 (p. 254) with paired LN converted to ATANH (1st version):

INT(1/(3+4*COS(x)+4*SIN(x)),x)=-1/SQRT(23)*ATANH(SQRT(23)*(COS(x~
)-SIN(x))/(8+3*COS(x)+3*SIN(x)))=1/SQRT(23)*LN((8+(3-SQRT(23))*C~
OS(x)+(3+SQRT(23))*SIN(x))/(3+4*COS(x)+4*SIN(x)))

- Example 63 (p. 268) expressed somewhat more compactly:

INT(1/TAN(5*x)^(1/3),x)=3/20*LN(1+TAN(5*x)^(2/3))-1/20*LN(1+TAN(~
5*x)^2)-SQRT(3)/10*ATAN((1-2*TAN(5*x)^(2/3))/SQRT(3))

- Example 70 (p. 272) with logarihms and arc tangents condensed:

INT(SIN(x)^7/SIN(2*x)^(7/2),x)=SIN(x)^5/(5*SIN(2*x)^(5/2))-SIN(x~
)/(4*SQRT(SIN(2*x)))+1/16*LN(COS(x)+SIN(x)+SQRT(SIN(2*x)))-1/16*~
ASIN(COS(x)-SIN(x))

- Example 71 (p. 272) ditto:

INT(COS(x)^7/SIN(2*x)^(7/2),x)=-COS(x)^5/(5*SIN(2*x)^(5/2))+COS(~
x)/(4*SQRT(SIN(2*x)))-1/16*LN(COS(x)+SIN(x)+SQRT(SIN(2*x)))-1/16~
*ASIN(COS(x)-SIN(x))

- Example 77 (p. 276) with piecewise constants absorbed into condensed
logarithms and arc tangents:

INT(SQRT(SIN(x)^5/COS(x)),x)=-COT(x)/2*SQRT(SIN(x)^5/COS(x))+3/(~
4*SQRT(2))*LN(SIN(x)+COS(x)-SQRT(2)*COT(x)/SIN(x)*SQRT(SIN(x)^5/~
COS(x)))+3/(4*SQRT(2))*ATAN((1-COT(x))/(SQRT(2)*SIN(x)^2)*SQRT(S~
IN(x)^5/COS(x)))

- Example 80 (p. 276) with LN, ASINH, and ATAN condensed in various
ways:

INT((COS(2*x)-SQRT(SIN(2*x)))/SQRT(SIN(x)*COS(x)^3),x)=-SIN(2*x)~
/SQRT(SIN(x)*COS(x)^3)-SQRT(2)*LN(SIN(x)+COS(x)-SQRT(2)*SEC(x)*S~
QRT(SIN(x)*COS(x)^3))-COS(x)*SQRT(SIN(2*x))/SQRT(SIN(x)*COS(x)^3~
)*ATANH(SIN(x))+COS(x)*SQRT(SIN(2*x))/SQRT(SIN(x)*COS(x)^3)*ASIN~
(SIN(x)-COS(x))

- Example 81 (p. 276) substantially reorganized and condensed (two
versions):

INT((SQRT(SIN(x)^3*COS(x))-2*SIN(2*x))/(SQRT(TAN(x))-SQRT(SIN(x)~
*COS(x)^3)),x)=4/SQRT(TAN(x))+4*SEC(x)*CSC(x)*SQRT(SIN(x)*COS(x)~
^3)+1/2*CSC(x)^2*SQRT(TAN(x))*SQRT(SIN(x)^3*COS(x))*LN(SIN(x))-1~
/4*CSC(x)^2*SQRT(TAN(x))*SQRT(SIN(x)^3*COS(x))*LN(1+COS(x)^2)+1/~
2*SEC(x)^2*CSC(x)^2*SQRT(SIN(x)^3*COS(x))*SQRT(SIN(x)*COS(x)^3)*~
LN(SIN(x))+1/4*SEC(x)^2*CSC(x)^2*SQRT(SIN(x)^3*COS(x))*SQRT(SIN(~
x)*COS(x)^3)*LN(1+COS(x)^2)-2^(1/4)*ACOTH((TAN(x)+SQRT(2))/(2^(3~
/4)*SQRT(TAN(x))))-2*SQRT(2)*ACOTH((SIN(x)+COS(x))/(SQRT(2)*SEC(~
x)*SQRT(SIN(x)*COS(x)^3)))+2^(1/4)*ACOTH((SIN(x)+SQRT(2)*COS(x))~
/(2^(3/4)*SEC(x)*SQRT(SIN(x)*COS(x)^3)))+2^(1/4)*ATAN((TAN(x)-SQ~
RT(2))/(2^(3/4)*SQRT(TAN(x))))+2*SQRT(2)*ATAN((SIN(x)-COS(x))/(S~
QRT(2)*SEC(x)*SQRT(SIN(x)*COS(x)^3)))-2^(1/4)*ATAN((SIN(x)-SQRT(~
2)*COS(x))/(2^(3/4)*SEC(x)*SQRT(SIN(x)*COS(x)^3)))=4/SQRT(TAN(x)~
)+4*SEC(x)*CSC(x)*SQRT(SIN(x)*COS(x)^3)+1/2*CSC(x)^2*SQRT(TAN(x)~
)*SQRT(SIN(x)^3*COS(x))*LN(SIN(x))-1/4*CSC(x)^2*SQRT(TAN(x))*SQR~
T(SIN(x)^3*COS(x))*LN(1+COS(x)^2)+1/2*SEC(x)^2*CSC(x)^2*SQRT(SIN~
(x)^3*COS(x))*SQRT(SIN(x)*COS(x)^3)*LN(SIN(x))+1/4*SEC(x)^2*CSC(~
x)^2*SQRT(SIN(x)^3*COS(x))*SQRT(SIN(x)*COS(x)^3)*LN(1+COS(x)^2)-~
2^(1/4)*LN(SIN(x)+SQRT(2)*COS(x)+2^(3/4)*COS(x)*SQRT(TAN(x)))-2*~
SQRT(2)*LN(SIN(x)+COS(x)+SQRT(2)*SEC(x)*SQRT(SIN(x)*COS(x)^3))+2~
^(1/4)*LN(SIN(x)+SQRT(2)*COS(x)+2^(3/4)*SEC(x)*SQRT(SIN(x)*COS(x~
)^3))+2^(1/4)*ATAN((TAN(x)-SQRT(2))/(2^(3/4)*SQRT(TAN(x))))+2*SQ~
RT(2)*ATAN((SIN(x)-COS(x))/(SQRT(2)*SEC(x)*SQRT(SIN(x)*COS(x)^3)~
))-2^(1/4)*ATAN((SIN(x)-SQRT(2)*COS(x))/(2^(3/4)*SEC(x)*SQRT(SIN~
(x)*COS(x)^3)))

- Example 108 (p. 303) involves a^4 +- b^4*CSC(x)^2 and should be
duplicated into 108a and 108b.

- Example 109 (p. 303) with LN and ATANH expressed more compactly:

INT((3*TAN(x)^2+SIN(x)^2*(1-3*SEC(x)^2)^(1/3))*TAN(x)/(COS(x)^2*~
(1-3*SEC(x)^2)^(5/6)*(1-SQRT(1-3*SEC(x)^2))),x)=1/(2*(1-SQRT(1-3~
*SEC(x)^2)))-1/4*(1-3*SEC(x)^2)^(2/3)-(1-3*SEC(x)^2)^(1/6)+1/4*L~
N(SEC(x)^2)+1/3*LN(1-SQRT(1-3*SEC(x)^2))-3/2*LN(1-(1-3*SEC(x)^2)~
^(1/6))+SQRT(3)*ATAN((1+2*(1-3*SEC(x)^2)^(1/6))/SQRT(3))

The radicand in this integrand is negative for real x.

- Example 110 (p. 303) with piecewise constants simplified and the arc

INT((2*TAN(x)^2-COS(2*x))/(COS(x)^2*(TAN(x)*TAN(2*x))^(3/2)),x)=~
3*TAN(x)/(4*SQRT(TAN(x)*TAN(2*x)))+TAN(x)/(2*(TAN(x)*TAN(2*x))^(~
3/2))+2*TAN(x)^3/(3*(TAN(x)*TAN(2*x))^(3/2))+2*ATANH(TAN(x)/SQRT~
(TAN(x)*TAN(2*x)))-11/(4*SQRT(2))*ATANH(SQRT(2)*TAN(x)/SQRT(TAN(~
x)*TAN(2*x)))

- Example 112 (p. 306) with paired LN converted to ATANH and
subsequently simplified:

INT((1+2*COS(x)^9)^(5/6)*TAN(x),x)=-2/15*(1+2*COS(x)^9)^(5/6)-1/~
9*ATANH(SQRT(1+2*COS(x)^9))+1/3*ATANH((1+2*COS(x)^9)^(1/6))-SQRT~
(3)/9*ATAN((1+2*(1+2*COS(x)^9)^(1/6))/SQRT(3))+SQRT(3)/9*ATAN((1~
-2*(1+2*COS(x)^9)^(1/6))/SQRT(3))

- Example 115 (p. 308) simplified according to Timofeev:

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

- Example 116 (p. 309) with algebraic prefactors of ATAN fused:

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

- Example 119 (p. 309) after some piecewise-constant acrobatics:

INT(SQRT(TAN(x)*TAN(2*x)),x)=-ATANH(TAN(x)/SQRT(TAN(x)*TAN(2*x)))

Note that SQRT(SEC(x)^3) has not been interpreted as SEC(x)^(3/2) in
Example 26 (p. 223), whereas SQRT(x^3) has been interpreted as x^(3/2)
in Example 1 of Chapter 4. Similar to what has been done in Example 44
above, the paired logarithms in the evaluations of example integrals 62,
64, 66 (p. 268), and 118 (p. 309) may be written as area hyperbolic
tangents, or split differently using:

(1 + TAN(x) + SQRT(2)*SQRT(TAN(x)))
* (1 + TAN(x) - SQRT(2)*SQRT(TAN(x)))
= SEC(x)^2

(3 + TAN(2*x) + SQRT(2)*SQRT(4 + 3*TAN(2*x)))
* (3 + TAN(2*x) - SQRT(2)*SQRT(4 + 3*TAN(2*x)))
= SEC(2*x)^2

(1 + SQRT(2*COS(x)^2 - 1) + SQRT(2)*(2*COS(x)^2 - 1)^(1/4))
* (1 + SQRT(2*COS(x)^2 - 1) - SQRT(2)*(2*COS(x)^2 - 1)^(1/4))
= 2*COS(x)^2

Also note that a systematic check of the evaluations for the Chapter 5
integrals against those in Timofeev's book is still to be made.

Martin.