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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: 1,245 Registered: 4/26/08
Re: An independent integration test suite
Posted: Dec 8, 2013 3:21 PM

clicliclic@freenet.de schrieb:
>
> [...]
>
> Perhaps other systems (and/or chapters) can be added here by and by.
> I am currently working on chapter 4 (132 integrals) in which I am
> expecially interested, but not expecting to finish this before some
> months have passed. I am not likely to enter the integrals from other
> chapters too (particularly not the remaining massive ones 2, 5, and
> 8).
>

This was written in April. I have now finished the digitization of
Examples from Chapter 4 of Timofeev's book; the algebraic integrals and
model solutions are appended below. Timofeev presents 132 Examples in
Chapter 4, of which the final group (numbers 123 - 132, p. 198-200) are
mere substitution recipes that had to be omitted here. With each Example
consisting of just a single integral, 122 entries result in total.

As in the digitization of other Chapters, nested powers (^r)^(1/s) were
interpreted as ^(r/s), with the exception of Example #1 (just for the
sake of variety). The correction of misprints was straightforward, apart
from Example #53 (p. 164) where the x dependence of the terms in
Timofeev's evaluation corresponds with the integrand denominator, while
their amplitudes are incompatible with the integrand numerator. No
simple correction of integrand and/or antiderivative appears feasible
here, so the integrand was adopted as printed while all the amplitudes
in the antiderivative were changed.

For the default setting of real variables, Derive 6.10 is found to fail
on Examples 13, 15, 17, 20-21, 35, 37-42, 49, 51, 53, 74, 76, 99-100,
107-108, 110, 112-114, and 117-122; additionally it cannot evaluate
Examples 14, 16-21, 23-27, and 46 if the integration variable (and the
occasional parameters) are declared complex.

Timofeev's book can hopefully still be found at:

<h t t p : / / e q w o r l d . i p m n e t . r u / r u / l i b r a r y /
m a t h e m a t i c s / c a l c u l u s . h t m>

<h t t p : / / e q w o r l d . i p m n e t . r u / r u / l i b r a r y /
b o o k s / T i m o f e e v 1 9 4 8 r u . d j v u>

Have fun!

Martin.

" Timofeev (1948) Chapter 4 Integration Examples "

" examples 1 - 10 (p. 115-117) ... "

INT(SQRT(x^3)*(1+x^2)*(2*SQRT(x)-x)^2,x)=x^2*SQRT(x^3)*(8/7-SQRT~
(x)+2/9*x+8/11*x^2-2/3*x^(5/2)+2/13*x^3)

INT((x^(3/2)-3*x^(3/5))^2*(4*x^(3/2)-1/3*x^(2/3)),x)=8/11*x^(11/~
2)-1/14*x^(14/3)-120/23*x^(23/5)+60/113*x^(113/30)+360/37*x^(37/~
10)-45/43*x^(43/15)

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

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

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

INT(1/((1+x)^(2/3)-SQRT(1+x)),x)=3*(1+x)^(1/3)+6*(1+x)^(1/6)+6*L~
N((1+x)^(1/6)-1)

INT((1+x^(1/4))^(1/3)/SQRT(x),x)=3/7*(4*SQRT(x)+x^(1/4)-3)*(1+x^~
(1/4))^(1/3)

INT(1/(x^3*(1+x)^(3/2)),x)=(15*x^2+5*x-2)/(4*x^2*SQRT(1+x))-15/4~
*ATANH(1/SQRT(1+x))

INT(1/(x^5*(1-x)^(7/2)),x)=1/(960*x^4*(1-x)^(5/2))*(45045*x^6-10~
5105*x^5+69069*x^4-6435*x^3-1430*x^2-520*x-240)-3003/64*ATANH(1/~
SQRT(1-x))

INT(1/(x^5*(x-1)^(2/3)),x)=(x-1)^(1/3)/(324*x^4)*(220*x^3+132*x^~
2+99*x+81)+55/243*LN((1+(x-1)^(1/3))^3/x)-110/(81*SQRT(3))*ATAN(~
(1-2*(x-1)^(1/3))/SQRT(3))

" examples 11 - 15 (p. 118-120) ... "

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

INT(SQRT((x-a)/(b-x))*x,x)=1/4*((a-3*b-2*x)*(b-x)*SQRT((x-a)/(b-~
x))+(a+3*b)*(a-b)*ATAN(1/SQRT((x-a)/(b-x))))

INT(SQRT(x-5)*SQRT(x+3)/((x-1)*(x^2-25)),x)=1/(3*SQRT(5))*ATANH(~
SQRT(x-5)/(SQRT(5)*SQRT(x+3)))+1/3*ATAN(SQRT(x-5)/SQRT(x+3))

INT(x^2*(1-x^2)^(1/4)*SQRT(1+x)/(SQRT(1-x)*(SQRT(1-x)-SQRT(1+x))~
),x)=1/48*(1-x)^(1/4)*(1+x)^(1/4)*((8*x^2+22*x+29)*SQRT(1-x)-(8*~
x^2+2*x-7)*SQRT(1+x))-SQRT(2)/8*ATANH(SQRT(2)*(1-x)^(1/4)*(1+x)^~
(1/4)/(SQRT(1-x)+SQRT(1+x)))+3*SQRT(2)/16*ATAN((SQRT(1-x)-SQRT(1~
+x))/(SQRT(2)*(1-x)^(1/4)*(1+x)^(1/4)))=1/48*(1-x)^(1/4)*(1+x)^(~
1/4)*((8*x^2+22*x+29)*SQRT(1-x)-(8*x^2+2*x-7)*SQRT(1+x))-SQRT(2)~
/8*LN(SQRT(1-x)+SQRT(1+x)+SQRT(2)*(1-x)^(1/4)*(1+x)^(1/4))+3*SQR~
T(2)/16*ATAN((SQRT(1-x)-SQRT(1+x))/(SQRT(2)*(1-x)^(1/4)*(1+x)^(1~
/4)))

INT(x*(1+x)^(2/3)*SQRT(1-x)/(SQRT(1+x)*(1-x)^(2/3)-(1+x)^(1/3)*(~
1-x)^(5/6)),x)=-1/12*(1-x)^(1/6)*(3*(3+x)*(1-x)^(5/6)+(10+3*x)*(~
1-x)^(2/3)*(1+x)^(1/6)+(1-3*x)*SQRT(1-x)*(1+x)^(1/3)-3*x*(1-x)^(~
1/3)*SQRT(1+x)-(1+3*x)*(1-x)^(1/6)*(1+x)^(2/3)-(2+3*x)*(1+x)^(5/~
6))+SQRT(3)/18*ATANH(SQRT(3)*(1-x)^(1/6)*(1+x)^(1/6)/((1-x)^(1/3~
)+(1+x)^(1/3)))+1/6*ATAN((1+x)^(1/6)/(1-x)^(1/6))-5/6*ATAN(((1-x~
)^(1/3)-(1+x)^(1/3))/((1-x)^(1/6)*(1+x)^(1/6)))-4*SQRT(3)/9*ATAN~
(((1-x)^(1/3)-2*(1+x)^(1/3))/(SQRT(3)*(1-x)^(1/3)))

" examples 16 - 27 (p. 127-128) ... "

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

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

INT(1/((x+1)^2*(x-1)^7)^(1/3),x)=3/16*((x+1)*(x-1)*(3*x-5)/((x+1~
)^2*(x-1)^7)^(1/3))

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

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

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

INT(1/(x^2-2*x-3)^(5/2),x)=(x-1)*(x^2-2*x-5)/(24*(x^2-2*x-3)^(3/~
2))

INT(1/SQRT(x^3-5*x^2+3*x+9),x)=-ATANH(SQRT(x^3-5*x^2+3*x+9)/(2*(~
x-3)))

INT(1/(x^3-5*x^2+3*x+9)^(3/2),x)=(15*x^2-70*x+43)/(256*(x-3)*SQR~
T(x^3-5*x^2+3*x+9))-15/512*ATANH(SQRT(x^3-5*x^2+3*x+9)/(2*(x-3)))

INT(1/(x^3-5*x^2+3*x+9)^(1/3),x)=(x+1)^(1/3)*(x-3)^(2/3)/(x^3-5*~
x^2+3*x+9)^(1/3)*(SQRT(3)*ATAN(((x+1)^(1/3)+2*(x-3)^(1/3))/(SQRT~
(3)*(x+1)^(1/3)))-3/2*LN((x+1)^(1/3)-(x-3)^(1/3)))

INT(1/(x^3-5*x^2+3*x+9)^(2/3),x)=-3*(x^3-5*x^2+3*x+9)^(1/3)/(4*(~
x-3))

INT(1/(x^3-5*x^2+3*x+9)^(4/3),x)=3*(9*x^2-42*x+29)/(320*(x-3)*(x~
^3-5*x^2+3*x+9)^(1/3))

" examples 28 - 42 (p. 143-146) ... "

INT(1/SQRT(4+3*x-2*x^2),x)=1/SQRT(2)*ASIN((4*x-3)/SQRT(41))

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

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

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

INT(1/((x^2+4)*SQRT(4*x^2+1)),x)=SQRT(15)/30*ATANH(SQRT(15)*x/(2~
*SQRT(4*x^2+1)))

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

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

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

INT(x/((x^2-1)*SQRT(x^2+2*x+4)),x)=-SQRT(3)/6*ATANH(SQRT(3)/SQRT~
(x^2+2*x+4))-SQRT(7)/14*ATANH((2*x+5)/(SQRT(7)*SQRT(x^2+2*x+4)))

INT(1/((x^3-8)*SQRT(x^2+2*x+5)),x)=1/12*ATANH(1/SQRT(x^2+2*x+5))~
-1/(12*SQRT(13))*ATANH((7+3*x)/(SQRT(13)*SQRT(x^2+2*x+5)))-SQRT(~
3)/12*ATAN((x+1)/(SQRT(3)*SQRT(x^2+2*x+5)))

INT(x/((x^2+x+4)*SQRT(4*x^2+4*x+5)),x)=-1/SQRT(165)*ATANH(SQRT(1~
1)*(2*x+1)/(SQRT(15)*SQRT(4*x^2+4*x+5)))-1/SQRT(11)*ATAN(SQRT(11~
)/SQRT(4*x^2+4*x+5))

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

INT((2*x+1)/((3*x^2+4*x+4)*SQRT(x^2+6*x-1)),x)=-1/(3*SQRT(7))*AC~
OTH(SQRT(7)*(1+x)/SQRT(x^2+6*x-1))-5/(6*SQRT(14))*ATAN(SQRT(7)*(~
2-x)/(SQRT(8)*SQRT(x^2+6*x-1)))

INT((a*x+b)/((5*x^2-18*x+17)*SQRT(10*x^2-22*x+13)),x)=(a+b)/(2*S~
QRT(35))*ATANH(SQRT(35)*(x-1)/(2*SQRT(10*x^2-22*x+13)))-(2*a+b)/~
SQRT(35)*ATAN(SQRT(35)*(2-x)/SQRT(10*x^2-22*x+13))

INT((x-2)/((5*x^2-18*x+17)*SQRT(10*x^2-22*x+13)),x)=-1/(2*SQRT(3~
5))*ATANH(SQRT(35)*(x-1)/(2*SQRT(10*x^2-22*x+13)))

" examples 43 - 84 (p. 163-167) ... "

INT(x^4*SQRT(5-x^2),x)=x/48*(8*x^4-10*x^2-75)*SQRT(5-x^2)+125/16~
*ASIN(x/SQRT(5))

INT(1/(x^6*SQRT(x^2+2)),x)=-(3-2*x^2+2*x^4)/(30*x^5)*SQRT(x^2+2)

INT(1/(2*x^2+3)^(7/2),x)=x*(32*x^4+120*x^2+135)/(405*(2*x^2+3)^(~
5/2))

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

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

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

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

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

INT((x^2+5)/((1+x^2)^2*SQRT(1-x^2)),x)=x*SQRT(1-x^2)/(1+x^2)+2*S~
QRT(2)*ATAN(SQRT(2)*x/SQRT(1-x^2))

INT((4*x-SQRT(1-x^2))/(5+SQRT(1-x^2)),x)=-x-4*SQRT(1-x^2)+20*LN(~
5+SQRT(1-x^2))-25/(2*SQRT(6))*ATAN(2*SQRT(6)*x/(1+5*SQRT(1-x^2))~
)+5*ASIN(x)

INT((2-SQRT(x^2+1))*x^2/(SQRT(x^2+1)*((x^2+1)^(3/2)-x^3+1)),x)=(~
16-3*x)/18*(x+SQRT(x^2+1))-7/54*LN(3*x^2+2*x+3)-41/54*ATANH(x/SQ~
RT(x^2+1))-7/27*ATANH((x-1)/(2*SQRT(x^2+1)))+4*SQRT(2)/27*ATAN((~
3*x+1)/SQRT(8))+4*SQRT(2)/27*ATAN((x+1)/(SQRT(2)*SQRT(x^2+1)))=(~
16-3*x)/18*(x+SQRT(x^2+1))-7/54*LN(3*x^2+2*x+3)-41/54*LN(x+SQRT(~
x^2+1))-7/27*ATANH((x-1)/(2*SQRT(x^2+1)))+4*SQRT(2)/27*ATAN((3*x~
+1)/SQRT(8))+4*SQRT(2)/27*ATAN((x+1)/(SQRT(2)*SQRT(x^2+1)))

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

INT(x^2*SQRT(2*r*x-x^2),x)=(6*x^3-2*r*x^2-5*r^2*x-15*r^3)/24*SQR~
T(2*r*x-x^2)+5/8*r^4*ATAN((x-r)/SQRT(2*r*x-x^2))

INT(x^3*SQRT(2*r*x-x^2),x)=(24*x^4-6*r*x^3-14*r^2*x^2-35*r^3*x-1~
05*r^4)/120*SQRT(2*r*x-x^2)+7/8*r^5*ATAN((x-r)/SQRT(2*r*x-x^2))

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

INT((3*x-2)/((x+1)^3*SQRT(2*x-x^2)),x)=-(4*x+9)/(6*(x+1)^2)*SQRT~
(2*x-x^2)-SQRT(3)/6*ATAN((2*x-1)/(SQRT(3)*SQRT(2*x-x^2)))

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

INT(x^3/SQRT(1+x+x^2),x)=(8*x^2-10*x-1)/24*SQRT(1+x+x^2)+7/16*AT~
ANH((1+2*x)/(2*SQRT(1+x+x^2)))=(8*x^2-10*x-1)/24*SQRT(1+x+x^2)+7~
/16*LN(1+2*x+2*SQRT(1+x+x^2))

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

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

INT(x^3/(1+x+x^2)^(3/2),x)=(3*x^2+7*x+5)/(3*SQRT(1+x+x^2))-3/2*A~
TANH((1+2*x)/(2*SQRT(1+x+x^2)))=(3*x^2+7*x+5)/(3*SQRT(1+x+x^2))-~
3/2*LN(1+2*x+2*SQRT(1+x+x^2))

INT(x^2*SQRT(1+x+x^2),x)=(48*x^3+8*x^2+14*x-37)/192*SQRT(1+x+x^2~
)+3/128*ATANH((1+2*x)/(2*SQRT(1+x+x^2)))=(48*x^3+8*x^2+14*x-37)/~
192*SQRT(1+x+x^2)+3/128*LN(1+2*x+2*SQRT(1+x+x^2))

INT((1+x+x^2)^(3/2),x)=(2*x+1)*(8*x^2+8*x+17)/64*SQRT(1+x+x^2)+2~
7/128*ATANH((1+2*x)/(2*SQRT(1+x+x^2)))=(2*x+1)*(8*x^2+8*x+17)/64~
*SQRT(1+x+x^2)+27/128*LN(1+2*x+2*SQRT(1+x+x^2))

INT((1+x+x^2)^(5/2),x)=(2*x+1)*(128*x^4+256*x^3+504*x^2+376*x+38~
3)/1536*SQRT(1+x+x^2)+135/1024*ATANH((1+2*x)/(2*SQRT(1+x+x^2)))=~
(2*x+1)*(128*x^4+256*x^3+504*x^2+376*x+383)/1536*SQRT(1+x+x^2)+1~
35/1024*LN(1+2*x+2*SQRT(1+x+x^2))

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

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

INT(1/(x^2*(1+x+x^2)^(3/2)),x)=-(5*x^2+7*x+3)/(3*x*SQRT(1+x+x^2)~
)+3/2*ATANH((2+x)/(2*SQRT(1+x+x^2)))

INT(1/(x^3*(1+x+x^2)^(3/2)),x)=(37*x^3+23*x^2+15*x-6)/(12*x^2*SQ~
RT(1+x+x^2))-3/8*ATANH((2+x)/(2*SQRT(1+x+x^2)))

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

INT(1/((x^3-x)*SQRT(x^2+2*x+4)),x)=1/2*ATANH((x+4)/(2*SQRT(x^2+2~
*x+4)))-SQRT(7)/14*ATANH((2*x+5)/(SQRT(7)*SQRT(x^2+2*x+4)))-SQRT~
(3)/6*ATANH(SQRT(3)/SQRT(x^2+2*x+4))

INT(SQRT(x^2+2*x+4)/(x-1)^2,x)=SQRT(x^2+2*x+4)/(1-x)+ATANH((x+1)~
/SQRT(x^2+2*x+4))-2/SQRT(7)*ATANH((2*x+5)/(SQRT(7)*SQRT(x^2+2*x+~
4)))=SQRT(x^2+2*x+4)/(1-x)+LN(x+1+SQRT(x^2+2*x+4))-2/SQRT(7)*ATA~
NH((2*x+5)/(SQRT(7)*SQRT(x^2+2*x+4)))

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

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

INT((x^4+1)/((x^2+x+1)*SQRT(x^2+x+2)),x)=(2*x-7)/4*SQRT(x^2+x+2)~
-1/8*ATANH((2*x+1)/(2*SQRT(x^2+x+2)))-ATANH(1/SQRT(x^2+x+2))+1/S~
QRT(3)*ATAN((2*x+1)/(SQRT(3)*SQRT(x^2+x+2)))=(2*x-7)/4*SQRT(x^2+~
x+2)-1/8*LN(2*x+1+2*SQRT(x^2+x+2))-ATANH(1/SQRT(x^2+x+2))+1/SQRT~
(3)*ATAN((2*x+1)/(SQRT(3)*SQRT(x^2+x+2)))

INT(1/(x^2+2*x+4)^(7/2),x)=(x+1)*(8*x^4+32*x^3+108*x^2+152*x+203~
)/(405*(x^2+2*x+4)^(5/2))

INT(1/(3*x^2+8*x+1)^(5/2),x)=(3*x+4)*(18*x^2+48*x-7)/(507*(3*x^2~
+8*x+1)^(3/2))

INT(1/(5+4*x-3*x^2)^(5/2),x)=(3*x-2)*(49+24*x-18*x^2)/(1083*(5+4~
*x-3*x^2)^(3/2))

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

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

INT(x^2/(2*x+1+2*SQRT(1+x+x^2)),x)=(48*x^3+8*x^2+14*x-37)/288*SQ~
RT(1+x+x^2)-x^3*(3*x+2)/18+1/64*ATANH((2*x+1)/(2*SQRT(1+x+x^2)))~
=(48*x^3+8*x^2+14*x-37)/288*SQRT(1+x+x^2)-x^3*(3*x+2)/18+1/64*LN~
(2*x+1+2*SQRT(1+x+x^2))

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

INT((x+1)/(SQRT(x^2+2*x+4)-SQRT(x^2+x+1)),x)=(x-3)/2*SQRT(x^2+2*~
x+4)+(2*x-7)/4*SQRT(x^2+x+1)-2*SQRT(7)*LN(2*x-1+SQRT(7)*SQRT(x^2~
+2*x+4))+2*SQRT(7)*LN(2*SQRT(7)*SQRT(x^2+x+1)-5*x-1)+11/2*ATANH(~
(x+1)/SQRT(x^2+2*x+4))+43/8*ATANH((2*x+1)/(2*SQRT(x^2+x+1)))=(x-~
3)/2*SQRT(x^2+2*x+4)+(2*x-7)/4*SQRT(x^2+x+1)-2*SQRT(7)*LN(2*x-1+~
SQRT(7)*SQRT(x^2+2*x+4))+2*SQRT(7)*LN(2*SQRT(7)*SQRT(x^2+x+1)-5*~
x-1)+11/2*LN(x+1+SQRT(x^2+2*x+4))+43/8*LN(2*x+1+2*SQRT(x^2+x+1))

" examples 85 - 100 (p. 177-178) ... "

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

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

INT((3*x-1)^(4/3)/x^2,x)=(9*x+1)*(3*x-1)^(1/3)/x+2*LN(x)-6*LN((3~
*x-1)^(1/3)+1)-4*SQRT(3)*ATAN((2*(3*x-1)^(1/3)-1)/SQRT(3))

INT((4-3*x)^(4/3)*x^2,x)=-(35*x^2+28*x+16)/455*(4-3*x)^(7/3)

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

INT(x/(3-2*SQRT(x))^(3/4),x)=-4/65*(5*x^(3/2)+10*x+24*SQRT(x)+14~
4)*(3-2*SQRT(x))^(1/4)

INT((2*SQRT(x)-1)^(5/4)/x^2,x)=(2-9*SQRT(x))*(2*SQRT(x)-1)^(1/4)~
/(2*x)+5*SQRT(2)/4*ATANH(SQRT(2)*(2*SQRT(x)-1)^(1/4)/(1+SQRT(2*S~
QRT(x)-1)))-5*SQRT(2)/4*ATAN((1-SQRT(2*SQRT(x)-1))/(SQRT(2)*(2*S~
QRT(x)-1)^(1/4)))

INT((x^7+1)^(1/3)*x^6,x)=3/28*(x^7+1)^(4/3)

INT(x^6/(x^7+1)^(5/3),x)=-3/(14*(x^7+1)^(2/3))

INT(1/(x*(2*x^7-27)^(2/3)),x)=1/42*LN((2*x^7-27)^(1/3)+3)-1/18*L~
N(x)+SQRT(3)/63*ATAN((2*(2*x^7-27)^(1/3)-3)/(3*SQRT(3)))

INT((x^7+1)^(2/3)/x^8,x)=-(x^7+1)^(2/3)/(7*x^7)+1/7*LN((x^7+1)^(~
1/3)-1)-1/3*LN(x)+2*SQRT(3)/21*ATAN((2*(x^7+1)^(1/3)+1)/SQRT(3))

INT((3+4*x^4)^(1/4)/x^2,x)=-(3+4*x^4)^(1/4)/x+1/SQRT(2)*ATANH(SQ~
RT(2)*x/(3+4*x^4)^(1/4))-1/SQRT(2)*ATAN(SQRT(2)*x/(3+4*x^4)^(1/4~
))

INT(x^2*(3+4*x^4)^(5/4),x)=x^3*(16*x^4+27)*(3+4*x^4)^(1/4)/32+45~
*SQRT(2)/256*ATANH(SQRT(2)*x/(3+4*x^4)^(1/4))-45*SQRT(2)/256*ATA~
N(SQRT(2)*x/(3+4*x^4)^(1/4))

INT(x^6*(3+4*x^4)^(1/4),x)=x^3*(16*x^4+3)*(3+4*x^4)^(1/4)/128-27~
*SQRT(2)/1024*ATANH(SQRT(2)*x/(3+4*x^4)^(1/4))+27*SQRT(2)/1024*A~
TAN(SQRT(2)*x/(3+4*x^4)^(1/4))

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

INT(SQRT(x*(1+x^(1/3))),x)=(384*x^(4/3)+48*x-56*x^(2/3)+70*x^(1/~
3)-105)/(640*x^(1/3))*SQRT(x*(1+x^(1/3)))+21/128*ATANH(x^(2/3)/S~
QRT(x*(1+x^(1/3))))

" examples 101 - 122 (p. 193-196) ... "

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

INT(x^9*SQRT(1+x^5+x^10),x)=(5+2*x^5+8*x^10)/120*SQRT(1+x^5+x^10~
)-3/80*ATANH((1+2*x^5)/(2*SQRT(1+x^5+x^10)))=(5+2*x^5+8*x^10)/12~
0*SQRT(1+x^5+x^10)-3/80*LN(1+2*x^5+2*SQRT(1+x^5+x^10))

INT(1/(x^5*SQRT(4+2*x^2+x^4)),x)=(3*x^2-4)/(64*x^4)*SQRT(4+2*x^2~
+x^4)+1/128*ATANH((x^2+4)/(2*SQRT(4+2*x^2+x^4)))

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

INT((x^4-3*x^2)^(3/5)*(2*x^3-3*x),x)=5/16*(x^4-3*x^2)^(8/5)

INT((3*x^8-2*x^5-x^2*(3*x^3-1)^(2/3))/(3*x^3-1)^(3/4),x)=4/243*(~
3*x^3-1)^(9/4)-4/33*(3*x^3-1)^(11/12)-4/27*(3*x^3-1)^(1/4)

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

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

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

INT((x^4+1)^(3/4)/(x^4+2)^2,x)=x/(8*(x^4+2))*(x^4+1)^(3/4)+3/(16~
*8^(1/4))*ATANH(x/(2^(1/4)*(x^4+1)^(1/4)))+3/(16*8^(1/4))*ATAN(x~
/(2^(1/4)*(x^4+1)^(1/4)))

INT((x^5-2)^2/((x^5+3)^3*(x^5+3)^(1/5)),x)=x*(97*x^10+462*x^5+11~
88)/(891*(x^5+3)^(11/5))

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

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

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

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

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

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

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

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

INT((1-x^2)/((1+2*a*x+x^2)*SQRT(1+2*a*x+2*b*x^2+2*a*x^3+x^4)),x)~
=1/SQRT(2*(1-b))*ATAN((a+2*(a^2-b+1)*x+a*x^2)/(SQRT(2*(1-b))*SQR~
T(1+2*a*x+2*b*x^2+2*a*x^3+x^4)))

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

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

" ... end of Timofeev Chapter 4 "

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