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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: Rubi 4.1 and the Timofeev test suite
Posted: Sep 29, 2013 4:49 PM

Albert Rich schrieb:
>
> I just posted revised Timofeev Chapter 9 test suite problems and
> results incorporating the above change at
>
> http://www.apmaths.uwo.ca/~arich/TimofeevChapter9TestResults.pdf
>
> which of course required reducing the grades for problem #49 from 2 to
> 1 for all the systems tested.

I have converted the 61 examples from Chapter 9 of Timofeev's book to
those in the book. Thus I discovered that two integrands (and their
antiderivatives) must be corrected: in example #45 a misprinted
(x^2-1)^(3/2) must be replaced by (x^2-1)^(5/2), and in example #46 your
ACSC(x)^2 must be replaced by ACSC(x)^4, as is actually printed.
Timofeev's evaluations correctly differentiate back to the revised
integrands.

For reasons of quality, uniformity, conciseness, and aesthetics, I also
propose to:
- replace 2*ATANH((1 + 2*x^2)) by LN(1 + 1/x^2) in examples #4, #33, #34
and #35 to get rid of the imaginary offset.
- replace 2*ATANH((1 - 2*x^2)) by LN(1 - 1/x^2) in example #42 for the
same reason and to get rid of bad branch cuts.
- express ACSC as ASEC in example #47 since this is the rule for all
other examples.
- change SQRT(2 - COSH(x)^2) to SQRT(1 - SINH(x)^2) in example #57
because the second radicand looks more natural.
- move the piecewise constant x/SQRT(x^2) into ATANH and simplify to
ATANH(1/SQRT(x^2)) in examples #40, #41 (and the old #45).
- move the piecewise constant x/SQRT(x^2) into ACOTH and simplify to
ATANH(1/SQRT(x^2)) in example #43.
- remove a constant term from the evaluation of example #36 as you see
fit.
- perhaps replace SQRT(x^2)/x by x/SQRT(x^2) in examples #37, #38, #43,
#47 and #53, as the latter form is used elsewhere. In the same vain,
simplify SQRT(x^2)/x^3 to 1/(x*SQRT(x^2)) in example #38, (simplify
SQRT(x^2)/x^2 to 1/SQRT(x^2) in the old example #46), simplify
SQRT(x^2)/x^3 to 1/(x*SQRT(x^2)) and SQRT(x^2)/x^5 to 1/(x^3*SQRT(x^2))
in example #47, and simplify SQRT(x^2)/x^2 to 1/SQRT(x^2) and
SQRT(x^2)/x^4 to 1/(x^2)^(3/2) in example #48.

The Derive file incorprating these changes is appended. For examples
involving dilogarithm functions, Timofeev's original evaluations in
terms of simpler integrals are included as well.

Derive 6.10 cannot evaluate examples #6, #46, #47, #48, #49 and #50 if
the integration variable is declared real (which is the default
setting); additionally it cannot evaluate examples #5, #38, #39, #40,
#41, #42, #44, #45, #53, #55, #56, #57 and #59 if the variable is
declared complex. These results are in accord with your score table.

Derive's implementation of the dilogarithm function being very
rudimentary, it also fails to evaluate any of the corresponding
examples. Also note that the double DILOG antiderivatives in the Derive
file rely on the Mathematica definition of the ATAN or ATANH functions,
which is incompatible with Derive's definitions.

Hurray! If the present transcription rate can be kept up, from some time
next year onward, the integrators of this world will thenceforth be
judged by their Timofeev Index!

Martin.

" Timofeev (1948) Chapter 9 Integration Examples "

" examples 1 - 6 (p. 391-392) ... "

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

INT(x^2*ASIN(x)^2,x)=-2/27*x^3-4/9*x+1/3*x^3*ASIN(x)^2+(2*x^2/9+~
4/9)*SQRT(1-x^2)*ASIN(x)

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

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

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

INT(ASEC(x)^4/x^5,x)=-3/(128*x^4)-45/(128*x^2)+(3/(16*x^4)+9/(16~
*x^2)-45/128)*ASEC(x)^2-(1/(4*x^4)-3/32)*ASEC(x)^4-(3/(32*x^3)+4~
5/(64*x))*SQRT(1-1/x^2)*ASEC(x)+(1/(4*x^3)+3/(8*x))*SQRT(1-1/x^2~
)*ASEC(x)^3

" examples 7 - 18 (p. 397-398) ... "

[INT(SQRT(1-x^2)*ASIN(x),x)=-1/4*x^2+1/4*ASIN(x)^2+x/2*SQRT(1-x^~
2)*ASIN(x),INT(SQRT(1-x^2)*ACOS(x),x)=1/4*x^2-1/4*ACOS(x)^2+x/2*~
SQRT(1-x^2)*ACOS(x)]

INT(x*SQRT(1-x^2)*ACOS(x),x)=-x/3+1/9*x^3-1/3*(1-x^2)^(3/2)*ACOS~
(x)

INT((1-x^2)^(3/2)*ASIN(x),x)=-5/16*x^2+1/16*x^4+3/16*ASIN(x)^2+x~
*(3/8*SQRT(1-x^2)+1/4*(1-x^2)^(3/2))*ASIN(x)

INT(x*(1-x^2)^(3/2)*ASIN(x),x)=x/5-2/15*x^3+1/25*x^5-1/5*(1-x^2)~
^(5/2)*ASIN(x)

INT(x^3*(1-x^2)^(3/2)*ACOS(x),x)=-2/35*x-1/105*x^3+8/175*x^5-1/4~
9*x^7-(1/5*(1-x^2)^(5/2)-1/7*(1-x^2)^(7/2))*ACOS(x)

INT((1-x^2)^(3/2)*ACOS(x)/x,x)=4/3*x-1/9*x^3+(SQRT(1-x^2)+1/3*(1~
-x^2)^(3/2))*ACOS(x)+INT(ACOS(x)/(x*SQRT(1-x^2)),x)=4/3*x-1/9*x^~
3+(SQRT(1-x^2)+1/3*(1-x^2)^(3/2))*ACOS(x)+2*ACOS(x)*ATANH(#i*x-S~
QRT(1-x^2))+#i*DILOG(1-#i*x+SQRT(1-x^2))-#i*DILOG(1+#i*x-SQRT(1-~
x^2))

INT((1-x^2)^(3/2)*ASIN(x)/x^6,x)=-1/(20*x^4)+1/(5*x^2)+1/5*LN(x)~
-1/(5*x^5)*(1-x^2)^(5/2)*ASIN(x)

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

INT(x^4*ASIN(x)/SQRT(1-x^2),x)=1/16*x^4+3/16*x^2+3/16*ASIN(x)^2-~
(1/4*x^3+3/8*x)*SQRT(1-x^2)*ASIN(x)

[INT(x*ASIN(x)/(1-x^2)^(3/2),x)=-ATANH(x)+ASIN(x)/SQRT(1-x^2),IN~
T(x*ACOS(x)/(1-x^2)^(3/2),x)=ATANH(x)+ACOS(x)/SQRT(1-x^2)]

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

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

" examples 19 - 22 (p. 401) ... "

INT(ASIN(x)/(x*(1-x^2)^(3/2)),x)=-ATANH(x)+ASIN(x)/SQRT(1-x^2)+I~
NT(ASIN(x)/(x*SQRT(1-x^2)),x)=-ATANH(x)+ASIN(x)/SQRT(1-x^2)-2*AS~
IN(x)*ATANH(#i*x+SQRT(1-x^2))+#i*DILOG(1+#i*x+SQRT(1-x^2))-#i*DI~
LOG(1-#i*x-SQRT(1-x^2))

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

INT(x*SQRT(1-x^2)*ACOS(x)^2,x)=4/9*SQRT(1-x^2)+2/27*(1-x^2)^(3/2~
)-(2/3*x-2/9*x^3)*ACOS(x)-1/3*(1-x^2)^(3/2)*ACOS(x)^2

INT(x^2*ASIN(x)^3/SQRT(1-x^2),x)=-3/8*x^2-(3/8-3/4*x^2)*ASIN(x)^~
2+1/8*ASIN(x)^4+3/4*x*SQRT(1-x^2)*ASIN(x)-x/2*SQRT(1-x^2)*ASIN(x~
)^3

" examples 23 - 26 (p. 404-405) ... "

INT(x*ATAN(x)/(1+x^2)^2,x)=x/(4*(1+x^2))-(1/(2*(1+x^2))-1/4)*ATA~
N(x)

INT(x*ATAN(x)/(1+x^2)^3,x)=x/(16*(1+x^2)^2)+3*x/(32*(1+x^2))-(1/~
(4*(1+x^2)^2)-3/32)*ATAN(x)

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

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

" examples 27 - 32 (p. 407-408) ... "

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

INT(x^3*ATAN(x)/(1+x^2)^2,x)=-x/(4*(1+x^2))+(1/(2*(1+x^2))-1/4)*~
ATAN(x)+INT(x*ATAN(x)/(1+x^2),x)=-x/(4*(1+x^2))+(1/(2*(1+x^2))-1~
/4)*ATAN(x)-#i/2*ATAN(x)^2-LN(2*#i/(#i-x))*ATAN(x)-#i/2*DILOG(2*~
#i/(#i-x))

INT(x^5*ATAN(x)/(1+x^2)^2,x)=x/(4*(1+x^2))-x/2-(1/(2*(1+x^2))-1/~
2*x^2-3/4)*ATAN(x)-2*INT(x*ATAN(x)/(1+x^2),x)=x/(4*(1+x^2))-x/2-~
(1/(2*(1+x^2))-1/2*x^2-3/4)*ATAN(x)+#i*ATAN(x)^2+2*LN(2*#i/(#i-x~
))*ATAN(x)+#i*DILOG(2*#i/(#i-x))

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

INT((1+x^2)*ATAN(x)/x^5,x)=-1/(12*x^3)-1/(4*x)-(1+x^2)^2/(4*x^4)~
*ATAN(x)

INT((1+x^2)^2*ATAN(x)/x^5,x)=-1/(12*x^3)-3/(4*x)-(1/(4*x^4)+1/x^~
2+3/4)*ATAN(x)+INT(ATAN(x)/x,x)=-1/(12*x^3)-3/(4*x)-(1/(4*x^4)+1~
/x^2+3/4)*ATAN(x)+#i/2*DILOG(1+#i*x)-#i/2*DILOG(1-#i*x)

" examples 33 - 36 (p. 409) ... "

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

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

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

INT(x^3*ATAN(x)^2/(1+x^2)^3,x)=(4+5*x^2)/(32*(1+x^2)^2)+x*(3+5*x~
^2)/(16*(1+x^2)^2)*ATAN(x)-(3+6*x^2-5*x^4)/(32*(1+x^2)^2)*ATAN(x~
)^2

" examples 37 - 43 (p. 412-414) ... "

INT(SQRT(x^2-1)*ASEC(x)/x^2,x)=-1/SQRT(x^2)-SQRT(x^2-1)/x*ASEC(x~
)+INT(ASEC(x)/SQRT(x^2-1),x)=-1/SQRT(x^2)-SQRT(x^2-1)/x*ASEC(x)-~
2*x/SQRT(x^2)*ASEC(x)*ATANH(#i/x-SQRT(1-1/x^2))+#i*x/SQRT(x^2)*D~
ILOG(1+#i/x-SQRT(1-1/x^2))-#i*x/SQRT(x^2)*DILOG(1-#i/x+SQRT(1-1/~
x^2))

INT((x^2-1)^(5/2)*ACSC(x)/x^3,x)=1/(4*x*SQRT(x^2))+x*SQRT(x^2)/6~
-7*x/(3*SQRT(x^2))*LN(x)+(1/3*(x^2-1)^(5/2)-5/3*(x^2-1)^(3/2)-5/~
2*SQRT(x^2-1))*(ACSC(x)/x^2)-5*x/(4*SQRT(x^2))*ACSC(x)^2

INT(SQRT(x^2-1)*ASEC(x)/x^4,x)=-1/(9*SQRT(x^2)^3)+1/(3*SQRT(x^2)~
)+(x^2-1)^(3/2)/(3*x^3)*ASEC(x)

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

INT(x^2*ASEC(x)/(x^2-1)^(5/2),x)=-SQRT(x^2)/(6*(x^2-1))-x^3*ASEC~
(x)/(3*(x^2-1)^(3/2))-1/6*ATANH(1/SQRT(x^2))

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

INT(x^6*ASEC(x)/(x^2-1)^(5/2),x)=(2-3*x^2)*SQRT(x^2)/(6*(x^2-1))~
+x*(15-20*x^2+3*x^4)/(6*(x^2-1)^(3/2))*ASEC(x)-13/6*ATANH(1/SQRT~
(x^2))+5/2*INT(ASEC(x)/SQRT(x^2-1),x)=(2-3*x^2)*SQRT(x^2)/(6*(x^~
2-1))+x*(15-20*x^2+3*x^4)/(6*(x^2-1)^(3/2))*ASEC(x)-13/6*ATANH(1~
/SQRT(x^2))-5*x/SQRT(x^2)*ASEC(x)*ATANH(#i/x-SQRT(x^2-1)/SQRT(x^~
2))+5*#i*x/(2*SQRT(x^2))*DILOG(1+#i/x-SQRT(x^2-1)/SQRT(x^2))-5*#~
i*x/(2*SQRT(x^2))*DILOG(1-#i/x+SQRT(x^2-1)/SQRT(x^2))

" examples 44 - 48 (p. 416-417) ... "

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

INT(ACSC(x)/(x^2*(x^2-1)^(5/2)),x)=SQRT(x^2)/(6*(x^2-1))-1/SQRT(~
x^2)-11/6*ATANH(1/SQRT(x^2))+(3-12*x^2+8*x^4)/(3*x*(x^2-1)^(3/2)~
)*ACSC(x)

INT(ACSC(x)^4/(x^2*SQRT(x^2-1)),x)=24*SQRT(x^2-1)/x+24/SQRT(x^2)~
*ACSC(x)-4/SQRT(x^2)*ACSC(x)^3-12*SQRT(x^2-1)/x*ACSC(x)^2+SQRT(x~
^2-1)/x*ACSC(x)^4

INT((x^2-1)^(3/2)*ASEC(x)^2/x^5,x)=-(2-17*x^2)*SQRT(x^2-1)/(64*x~
^4)+(8-40*x^2+17*x^4)/(64*x^3*SQRT(x^2))*ASEC(x)+(2-5*x^2)*SQRT(~
x^2-1)/(8*x^4)*ASEC(x)^2+x/(8*SQRT(x^2))*ASEC(x)^3

INT(SQRT(x^2-1)*ASEC(x)^3/x^4,x)=2*(1-21*x^2)/(27*SQRT(x^2)^3)-(~
1-3*x^2)/(3*SQRT(x^2)^3)*ASEC(x)^2+2*(1-7*x^2)*SQRT(x^2-1)/(9*x^~
3)*ASEC(x)+(x^2-1)^(3/2)/(3*x^3)*ASEC(x)^3

" examples 49 - 56 (p. 422) ... "

INT(ASIN(SQRT((x-a)/(x+a))),x)=(x+a)*(-SQRT(2*a/(x+a))*SQRT((x-a~
)/(x+a))+ASIN(SQRT((x-a)/(x+a))))

INT(ATAN(SQRT((x-a)/(x+a))),x)=x*ATAN(SQRT((x-a)/(x+a)))-a*ATANH~
(SQRT((x-a)/(x+a)))

INT(ATAN(x)/(1+x)^3,x)=-1/(4*(1+x))+1/4*LN(1+x)-1/8*LN(1+x^2)-AT~
AN(x)/(2*(1+x)^2)

INT(ATAN(x-a)/(x+a),x)=LN(x+a)*ATAN(x-a)-INT(LN(x+a)/(1+(x-a)^2)~
,x)=#i/2*LN(1-#i*(x-a))*LN((x+a)/(2*a-#i))-#i/2*LN(1+#i*(x-a))*L~
N((x+a)/(2*a+#i))+#i/2*DILOG((x+a)/(2*a-#i))-#i/2*DILOG((x+a)/(2~
*a+#i))

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

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

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

INT((x-1)^(5/2)*ACSC(x),x)=x/SQRT(x^2)*(4/35*(x+1)^2-20/21*(x+1)~
+4)*(SQRT(x^2-1)/SQRT(x-1))+4/7*(x/SQRT(x^2))*ATANH(SQRT(x^2-1)/~
SQRT(x-1))+2/7*(x-1)^(7/2)*ACSC(x)

" examples 57 - 59 (p. 427) ... "

INT(SECH(x)^4*ASIN(SINH(x)),x)=1/6*SECH(x)*SQRT(1-SINH(x)^2)-2/3~
*ASIN(COSH(x)/SQRT(2))+(TANH(x)-1/3*TANH(x)^3)*ASIN(SINH(x))

INT(COSH(x)*ACOT(COSH(x))/SINH(x)^4,x)=1/6*COTH(x)+SQRT(2)/12*AT~
ANH(TANH(x)/SQRT(2))-1/3*CSCH(x)^3*ACOT(COSH(x))

INT(#e^x*ASIN(TANH(x)),x)=#e^x*ASIN(TANH(x))-COSH(x)*SQRT(SECH(x~
)^2)*LN(1+#e^(2*x))

" ... end of Timofeev Chapter 9 "

Date Subject Author
2/24/13 clicliclic@freenet.de
3/19/13 clicliclic@freenet.de
3/21/13 Waldek Hebisch
3/22/13 clicliclic@freenet.de
3/26/13 Waldek Hebisch
3/26/13 clicliclic@freenet.de
4/20/13 clicliclic@freenet.de
4/20/13 Nasser Abbasi
4/20/13 Rouben Rostamian
4/20/13 clicliclic@freenet.de
4/20/13 Rouben Rostamian
4/20/13 Axel Vogt
4/20/13 clicliclic@freenet.de
4/20/13 Axel Vogt
4/21/13 Axel Vogt
4/21/13 clicliclic@freenet.de
4/21/13 Waldek Hebisch
4/22/13 clicliclic@freenet.de
4/22/13 Axel Vogt
4/22/13 clicliclic@freenet.de
4/23/13 Waldek Hebisch
4/24/13 clicliclic@freenet.de
4/25/13 Waldek Hebisch
4/26/13 clicliclic@freenet.de
4/27/13 Waldek Hebisch
4/24/13 Richard Fateman
4/24/13 clicliclic@freenet.de
4/25/13 Richard Fateman
4/26/13 clicliclic@freenet.de
4/26/13 Axel Vogt
4/27/13 clicliclic@freenet.de
4/25/13 Waldek Hebisch
4/25/13 Peter Pein
4/25/13 Nasser Abbasi
4/26/13 Peter Pein
4/26/13 clicliclic@freenet.de
4/26/13 Peter Pein
4/26/13 clicliclic@freenet.de
4/26/13 Richard Fateman
4/27/13 clicliclic@freenet.de
4/27/13 Richard Fateman
6/30/13 clicliclic@freenet.de
6/30/13 Axel Vogt
7/1/13 clicliclic@freenet.de
7/1/13 Axel Vogt
7/1/13 Waldek Hebisch
7/2/13 clicliclic@freenet.de
7/2/13 clicliclic@freenet.de
7/2/13 clicliclic@freenet.de
7/2/13 Nasser Abbasi
7/2/13 Nasser Abbasi
7/4/13 clicliclic@freenet.de
7/4/13 Nasser Abbasi
7/4/13 Nasser Abbasi
7/5/13 clicliclic@freenet.de
7/5/13 Nasser Abbasi
7/9/13 clicliclic@freenet.de
7/10/13 Nasser Abbasi
7/10/13 Richard Fateman
7/10/13 Nasser Abbasi
7/10/13 clicliclic@freenet.de
8/6/13 clicliclic@freenet.de
9/15/13 Albert D. Rich
9/15/13 clicliclic@freenet.de
9/15/13 clicliclic@freenet.de
9/21/13 Albert D. Rich
9/21/13 clicliclic@freenet.de
9/22/13 daly@axiom-developer.org
9/24/13 daly@axiom-developer.org
9/30/13 daly@axiom-developer.org
9/22/13 Albert D. Rich
9/25/13 Albert D. Rich
9/25/13 Albert D. Rich
9/25/13 clicliclic@freenet.de
9/25/13 Albert D. Rich
9/26/13 Albert D. Rich
9/26/13 clicliclic@freenet.de
9/26/13 Albert D. Rich
9/29/13 clicliclic@freenet.de
10/1/13 Albert D. Rich
10/1/13 clicliclic@freenet.de
10/1/13 Albert D. Rich
10/5/13 clicliclic@freenet.de
10/5/13 Albert D. Rich
10/6/13 clicliclic@freenet.de
10/10/13 Albert D. Rich
10/10/13 Nasser Abbasi
10/11/13 clicliclic@freenet.de
11/6/13 Albert D. Rich
11/6/13 Nasser Abbasi
11/7/13 did
11/7/13 clicliclic@freenet.de
11/7/13 clicliclic@freenet.de
11/7/13 Albert D. Rich
11/12/13 clicliclic@freenet.de
11/12/13 Albert D. Rich
11/13/13 clicliclic@freenet.de
11/13/13 Albert D. Rich
11/14/13 clicliclic@freenet.de
11/14/13 Albert D. Rich
11/15/13 clicliclic@freenet.de
11/15/13 Albert D. Rich
11/16/13 clicliclic@freenet.de
11/16/13 clicliclic@freenet.de
11/21/13 Albert D. Rich
11/21/13 clicliclic@freenet.de
11/21/13 Nasser Abbasi
11/21/13 Albert D. Rich
11/21/13 Albert D. Rich
11/22/13 clicliclic@freenet.de
11/14/13 Albert D. Rich
11/15/13 clicliclic@freenet.de
11/15/13 Nasser Abbasi
11/16/13 clicliclic@freenet.de
11/16/13 Nasser Abbasi
11/7/13 did
11/7/13 clicliclic@freenet.de
4/20/13 Richard Fateman
4/21/13 clicliclic@freenet.de
4/20/13 Axel Vogt
4/20/13 clicliclic@freenet.de
4/20/13 Waldek Hebisch
4/21/13 G. A. Edgar
12/8/13 clicliclic@freenet.de
10/5/13 Albert D. Rich
10/6/13 clicliclic@freenet.de