The Charlwood Fifty

[Albert Rich]

There has been much discussion in recent sci.math.symbolic posts about the performance of various systems on 10 integration problems from Kevin Charlwood's 2008 article "Integration on Computer Algebra Systems". An appendix to his article includes 40 more problems.

It seems to me if these problems are going be used as a test-suite, we should start by trying to reach consensus as to what the best answer is. To that end I have posted a pdf file at

http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf

showing the 50 integrals and the best antiderivatives I have found so far. This was done with the help of the article as well as Mathematica, Maple, Rubi and Derive.

If you should find substantially better ones and would like to improve the test-suite, please post them on sci.math.symbolic so I can include them in the Charlwood Fifty. Substantial improvements include significantly simpler and more compact, involve elementary rather than special functions, involve special rather than hypergeometric functions, real rather than complex, continuous rather than discontinuous, etc.

Albert

[Nasser M. Abbasi]

fyi;

I've run the first 10 integrals only on M 9.01, Rubi4 and Maple 17.

Used your PDF above as the correct/expected result and came with a draft
result table. Few of the results of CAS are too large to
know by looking at the result if they are the same as what you showed,
so this needs additional simplification and other means to find out.

I've put a "?" next to those until it is decide for sure if these
are "correct" or not. Most likely they are correct but not optimal.

All the notebooks I've used are here also:

http://12000.org/my_notes/ten_hard_integrals/report.htm

Here is current result, any one please feel free to correct, and
I will update it

integral---->1 2 3 4 5 6 7 8 9 10
==================================================
M 9.01------>OK OK ? OK No ? No ? ? OK====> 4/10
Rubi4------->No OK ? No ? Ok ? OK OK OK====> 5/10
Male 17----->Ok Ok ? ? ? OK OK ? No No====> 4/10

--Nasser

[clicl...@freenet.de]

Albert Rich schrieb:

Hello,

here are my antiderivatives for problems #1 to #10, as promised. I think
they are as compact, continuous, real, and elementary as one could wish
them to be:

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

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

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

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

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

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

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

INT(SQRT(TAN(x)^2 + 2*TAN(x) + 2), x) =
= ASINH(TAN(x) + 1) + SQRT(2*SQRT(5) + 2)/2
*ATAN(SQRT(2)*(TAN(x)*SQRT(SQRT(5) + 1) - SQRT(SQRT(5) - 1))
/(2*SQRT(TAN(x)^2 + 2*TAN(x) + 2))) - SQRT(2*SQRT(5) - 2)/2
*ATANH(SQRT(2)*(TAN(x)*SQRT(SQRT(5) - 1) + SQRT(SQRT(5) + 1))
/(2*SQRT(TAN(x)^2 + 2*TAN(x) + 2)))

INT(SIN(x)*ATAN(SQRT(SEC(x) - 1)), x) =
= COS(x)*(1/2*SQRT(SEC(x) - 1) - ATAN(SQRT(SEC(x) - 1)))
- 1/2*ATAN(1/SQRT(SEC(x) - 1))

INT(x^3*EXP(ASIN(x))/SQRT(1 - x^2), x) =
= EXP(ASIN(x))/10*(x^3 + 3*x - 3*(x^2 + 1)*SQRT(1 - x^2))

Many of them agree (or almost agree) with your solutions. The result for
#5 can be shortened somewhat if continuity is of no concern:

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

I have just the time to add a few observations on your solutions for
Charlwood's integrals #11 to #50: The integration variable for problems
#27, #30, #31, #32, #33, #34, #35 is theta, not x. Problems #28 and #29
are (of course) pseudo-elliptic:

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

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

When these two integrands and their antiderivatives are averaged, one
obtains a pseudo-elliptic integral already known to Euler:

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

See the postscript to the Lettre de Fuss � Condorcet of May 1778,
reprinted in the Bulletin des Sciences Math�matiques et Astronomiques
(2e s�rie), tome 3, no 1 (1879), p. 225-227:

<http://www.numdam.org/item?id=BSMA_1879_2_3_1_225_0>

Martin.

[Albert Rich]

I just posted a revised pdf file of the Charlwood Fifty integration test-suite at

http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf

that incorporates the changes Martin suggested. Please disregard the previous version of the file. Also posted is a Mathematica package file of the test-suite in machine readable form at

http://www.apmaths.uwo.ca/~arich/CharlwoodProblems.m

Not yet found are elementary antiderivatives for problems 45 and 49, and several other antiderivatives are obviously not optimal...

Albert

[Waldek Hebisch]

AFAICS 49 leads to genuine elliptic integral, so no elementary
antiderivative. For 45 FriCAS produces the monster below.
This seem the be expansion of single log term with respect
to eight roots of -4:

+---------+
| +-+
4+-+ |2\|2 - 4
4\|2 |---------
| +-+
\|2\|2 - 3
*
atan
+-+ +-+ +-+
((2\|2 + 2)sin(x) + (- 2\|2 - 2)cos(x) - 2\|2 - 2)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ 4|-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-----------+
+-+ 4+-+ +-+ 4+-+ |2sin(x) - 2
((- 2\|2 - 4)\|2 cos(x) + (- 2\|2 - 4)\|2 ) |-----------
\| cos(x)
/
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 + 4 |2sin(x) - 2
((\|2 + 2)cos(x) + \|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 + 3
*
ROOT
+-+ 3 +-+ 2
(48\|2 + 68)cos(x) + (48\|2 + 68)cos(x)
+
+-+ +-+
(- 192\|2 - 272)cos(x) - 192\|2 - 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 - 204)cos(x) + (- 144\|2 - 204)cos(x)
+
+-+ +-+
(192\|2 + 272)cos(x) + 192\|2 + 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(- 29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 2
(58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(116\|2 + 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4
(29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 3
(87\|2 + 123)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 - 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-+ 4+-+ 3
(- 24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ 2
(- 24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 + 136)\|2 cos(x) + (96\|2 + 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ 2
(72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 - 136)\|2 cos(x) + (- 96\|2 - 136)\|2
*
+---------+
| +-+
|2\|2 + 4
|---------
| +-+
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3
(48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 2
(48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 - 272)\|2 cos(x) + (- 192\|2 - 272)\|2
*
sin(x)
+
+-+ 4+-+2 3
(- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 2
(- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 + 272)\|2 cos(x) + (192\|2 + 272)\|2
/
+-+ 3 +-+ 2
(12\|2 + 17)cos(x) + (12\|2 + 17)cos(x)
+
+-+ +-+
(- 48\|2 - 68)cos(x) - 48\|2 - 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 - 51)cos(x) + (- 36\|2 - 51)cos(x)
+
+-+ +-+
(48\|2 + 68)cos(x) + 48\|2 + 68
+
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 + 4 |2sin(x) - 2
((2\|2 + 2)cos(x) + 2\|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-----------+
+-+4+-+ +-+4+-+ |2sin(x) - 2
(- 2\|2 \|2 cos(x) - 2\|2 \|2 ) |-----------
\| cos(x)
+
+---------+
| +-+
4+-+ |2\|2 - 4
4\|2 |---------
| +-+
\|2\|2 - 3
*
atan
+-+ +-+ +-+
((2\|2 + 2)sin(x) + (- 2\|2 - 2)cos(x) - 2\|2 - 2)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ 4|-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-----------+
+-+ 4+-+ +-+ 4+-+ |2sin(x) - 2
((- 2\|2 - 4)\|2 cos(x) + (- 2\|2 - 4)\|2 ) |-----------
\| cos(x)
/
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 + 4 |2sin(x) - 2
((\|2 + 2)cos(x) + \|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 + 3
*
ROOT
+-+ 3 +-+ 2
(48\|2 + 68)cos(x) + (48\|2 + 68)cos(x)
+
+-+ +-+
(- 192\|2 - 272)cos(x) - 192\|2 - 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 - 204)cos(x) + (- 144\|2 - 204)cos(x)
+
+-+ +-+
(192\|2 + 272)cos(x) + 192\|2 + 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(- 29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 2
(58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(116\|2 + 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4
(29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 3
(87\|2 + 123)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 - 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-+ 4+-+ 3
(24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ 2
(24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 - 136)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 - 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(- 72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ 2
(- 72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 + 136)\|2 cos(x) + (96\|2 + 136)\|2
*
+---------+
| +-+
|2\|2 + 4
|---------
| +-+
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3
(48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 2
(48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 - 272)\|2 cos(x) + (- 192\|2 - 272)\|2
*
sin(x)
+
+-+ 4+-+2 3
(- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 2
(- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 + 272)\|2 cos(x) + (192\|2 + 272)\|2
/
+-+ 3 +-+ 2
(12\|2 + 17)cos(x) + (12\|2 + 17)cos(x)
+
+-+ +-+
(- 48\|2 - 68)cos(x) - 48\|2 - 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 - 51)cos(x) + (- 36\|2 - 51)cos(x)
+
+-+ +-+
(48\|2 + 68)cos(x) + 48\|2 + 68
+
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 + 4 |2sin(x) - 2
((2\|2 + 2)cos(x) + 2\|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-----------+
+-+4+-+ +-+4+-+ |2sin(x) - 2
(2\|2 \|2 cos(x) + 2\|2 \|2 ) |-----------
\| cos(x)
+
+---------+
| +-+
+-+ 4+-+ |2\|2 + 4
(- 8\|2 - 12)\|2 |---------
| +-+
\|2\|2 + 3
*
atan
+-+ +-+ +-+
((2\|2 - 2)sin(x) + (- 2\|2 + 2)cos(x) - 2\|2 + 2)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ 4|-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-----------+
+-+ 4+-+ +-+ 4+-+ |2sin(x) - 2
((- 2\|2 + 4)\|2 cos(x) + (- 2\|2 + 4)\|2 ) |-----------
\| cos(x)
/
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 - 4 |2sin(x) - 2
((\|2 - 2)cos(x) + \|2 - 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 - 3
*
ROOT
+-+ 3 +-+ 2
(48\|2 - 68)cos(x) + (48\|2 - 68)cos(x)
+
+-+ +-+
(- 192\|2 + 272)cos(x) - 192\|2 + 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 + 204)cos(x) + (- 144\|2 + 204)cos(x)
+
+-+ +-+
(192\|2 - 272)cos(x) + 192\|2 - 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(- 29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 2
(58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(116\|2 - 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4
(29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 3
(87\|2 - 123)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 + 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-+ 4+-+ 3
(- 24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ 2
(- 24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 - 136)\|2 cos(x) + (96\|2 - 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ 2
(72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 + 136)\|2 cos(x) + (- 96\|2 + 136)\|2
*
+---------+
| +-+
|2\|2 - 4
|---------
| +-+
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3
(48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 2
(48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 + 272)\|2 cos(x) + (- 192\|2 + 272)\|2
*
sin(x)
+
+-+ 4+-+2 3
(- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 2
(- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 - 272)\|2 cos(x) + (192\|2 - 272)\|2
/
+-+ 3 +-+ 2
(12\|2 - 17)cos(x) + (12\|2 - 17)cos(x)
+
+-+ +-+
(- 48\|2 + 68)cos(x) - 48\|2 + 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 + 51)cos(x) + (- 36\|2 + 51)cos(x)
+
+-+ +-+
(48\|2 - 68)cos(x) + 48\|2 - 68
+
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 - 4 |2sin(x) - 2
((- 2\|2 + 2)cos(x) - 2\|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-----------+
+-+4+-+ +-+4+-+ |2sin(x) - 2
(2\|2 \|2 cos(x) + 2\|2 \|2 ) |-----------
\| cos(x)
+
+---------+
| +-+
+-+ 4+-+ |2\|2 + 4
(- 8\|2 - 12)\|2 |---------
| +-+
\|2\|2 + 3
*
atan
+-+ +-+ +-+
((2\|2 - 2)sin(x) + (- 2\|2 + 2)cos(x) - 2\|2 + 2)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ 4|-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-----------+
+-+ 4+-+ +-+ 4+-+ |2sin(x) - 2
((- 2\|2 + 4)\|2 cos(x) + (- 2\|2 + 4)\|2 ) |-----------
\| cos(x)
/
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 - 4 |2sin(x) - 2
((\|2 - 2)cos(x) + \|2 - 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 - 3
*
ROOT
+-+ 3 +-+ 2
(48\|2 - 68)cos(x) + (48\|2 - 68)cos(x)
+
+-+ +-+
(- 192\|2 + 272)cos(x) - 192\|2 + 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 + 204)cos(x) + (- 144\|2 + 204)cos(x)
+
+-+ +-+
(192\|2 - 272)cos(x) + 192\|2 - 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(- 29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 2
(58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(116\|2 - 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4
(29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 3
(87\|2 - 123)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 + 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-+ 4+-+ 3
(24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ 2
(24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 + 136)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 + 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(- 72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ 2
(- 72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 - 136)\|2 cos(x) + (96\|2 - 136)\|2
*
+---------+
| +-+
|2\|2 - 4
|---------
| +-+
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3
(48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 2
(48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 + 272)\|2 cos(x) + (- 192\|2 + 272)\|2
*
sin(x)
+
+-+ 4+-+2 3
(- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 2
(- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 - 272)\|2 cos(x) + (192\|2 - 272)\|2
/
+-+ 3 +-+ 2
(12\|2 - 17)cos(x) + (12\|2 - 17)cos(x)
+
+-+ +-+
(- 48\|2 + 68)cos(x) - 48\|2 + 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 + 51)cos(x) + (- 36\|2 + 51)cos(x)
+
+-+ +-+
(48\|2 - 68)cos(x) + 48\|2 - 68
+
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 - 4 |2sin(x) - 2
((- 2\|2 + 2)cos(x) - 2\|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-----------+
+-+4+-+ +-+4+-+ |2sin(x) - 2
(- 2\|2 \|2 cos(x) - 2\|2 \|2 ) |-----------
\| cos(x)
+
+---------+
| +-+
+-+ 4+-+ |2\|2 + 4
(8\|2 + 12)\|2 |---------
| +-+
\|2\|2 + 3
*
atan
+-+ +-+ +-+
((2\|2 - 2)sin(x) + (- 2\|2 + 2)cos(x) - 2\|2 + 2)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ 4|-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-----------+
+-+ 4+-+ +-+ 4+-+ |2sin(x) - 2
((2\|2 - 4)\|2 cos(x) + (2\|2 - 4)\|2 ) |-----------
\| cos(x)
/
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 - 4 |2sin(x) - 2
((\|2 - 2)cos(x) + \|2 - 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 - 3
*
ROOT
+-+ 3 +-+ 2
(48\|2 - 68)cos(x) + (48\|2 - 68)cos(x)
+
+-+ +-+
(- 192\|2 + 272)cos(x) - 192\|2 + 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 + 204)cos(x) + (- 144\|2 + 204)cos(x)
+
+-+ +-+
(192\|2 - 272)cos(x) + 192\|2 - 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 + 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4
(- 29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 3
(- 87\|2 + 123)\|2 cos(x)
+
+-+ 4+-+ 2
(58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(116\|2 - 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-+ 4+-+ 3
(- 24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ 2
(- 24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 - 136)\|2 cos(x) + (96\|2 - 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ 2
(72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 + 136)\|2 cos(x) + (- 96\|2 + 136)\|2
*
+---------+
| +-+
|2\|2 - 4
|---------
| +-+
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3
(48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 2
(48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 + 272)\|2 cos(x) + (- 192\|2 + 272)\|2
*
sin(x)
+
+-+ 4+-+2 3
(- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 2
(- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 - 272)\|2 cos(x) + (192\|2 - 272)\|2
/
+-+ 3 +-+ 2
(12\|2 - 17)cos(x) + (12\|2 - 17)cos(x)
+
+-+ +-+
(- 48\|2 + 68)cos(x) - 48\|2 + 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 + 51)cos(x) + (- 36\|2 + 51)cos(x)
+
+-+ +-+
(48\|2 - 68)cos(x) + 48\|2 - 68
+
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 - 4 |2sin(x) - 2
((- 2\|2 + 2)cos(x) - 2\|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-----------+
+-+4+-+ +-+4+-+ |2sin(x) - 2
(2\|2 \|2 cos(x) + 2\|2 \|2 ) |-----------
\| cos(x)
+
+---------+
| +-+
+-+ 4+-+ |2\|2 + 4
(8\|2 + 12)\|2 |---------
| +-+
\|2\|2 + 3
*
atan
+-+ +-+ +-+
((2\|2 - 2)sin(x) + (- 2\|2 + 2)cos(x) - 2\|2 + 2)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ 4|-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-----------+
+-+ 4+-+ +-+ 4+-+ |2sin(x) - 2
((2\|2 - 4)\|2 cos(x) + (2\|2 - 4)\|2 ) |-----------
\| cos(x)
/
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 - 4 |2sin(x) - 2
((\|2 - 2)cos(x) + \|2 - 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 - 3
*
ROOT
+-+ 3 +-+ 2
(48\|2 - 68)cos(x) + (48\|2 - 68)cos(x)
+
+-+ +-+
(- 192\|2 + 272)cos(x) - 192\|2 + 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 + 204)cos(x) + (- 144\|2 + 204)cos(x)
+
+-+ +-+
(192\|2 - 272)cos(x) + 192\|2 - 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 + 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4
(- 29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 3
(- 87\|2 + 123)\|2 cos(x)
+
+-+ 4+-+ 2
(58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(116\|2 - 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-+ 4+-+ 3
(24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ 2
(24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 + 136)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 + 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(- 72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ 2
(- 72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 - 136)\|2 cos(x) + (96\|2 - 136)\|2
*
+---------+
| +-+
|2\|2 - 4
|---------
| +-+
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3
(48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 2
(48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 + 272)\|2 cos(x) + (- 192\|2 + 272)\|2
*
sin(x)
+
+-+ 4+-+2 3
(- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 2
(- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 - 272)\|2 cos(x) + (192\|2 - 272)\|2
/
+-+ 3 +-+ 2
(12\|2 - 17)cos(x) + (12\|2 - 17)cos(x)
+
+-+ +-+
(- 48\|2 + 68)cos(x) - 48\|2 + 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 + 51)cos(x) + (- 36\|2 + 51)cos(x)
+
+-+ +-+
(48\|2 - 68)cos(x) + 48\|2 - 68
+
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 - 4 |2sin(x) - 2
((- 2\|2 + 2)cos(x) - 2\|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-----------+
+-+4+-+ +-+4+-+ |2sin(x) - 2
(- 2\|2 \|2 cos(x) - 2\|2 \|2 ) |-----------
\| cos(x)
+
-
+---------+
| +-+
4+-+ |2\|2 - 4
4\|2 |---------
| +-+
\|2\|2 - 3
*
atan
+-+ +-+ +-+
((2\|2 + 2)sin(x) + (- 2\|2 - 2)cos(x) - 2\|2 - 2)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ 4|-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-----------+
+-+ 4+-+ +-+ 4+-+ |2sin(x) - 2
((2\|2 + 4)\|2 cos(x) + (2\|2 + 4)\|2 ) |-----------
\| cos(x)
/
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 + 4 |2sin(x) - 2
((\|2 + 2)cos(x) + \|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 + 3
*
ROOT
+-+ 3 +-+ 2
(48\|2 + 68)cos(x) + (48\|2 + 68)cos(x)
+
+-+ +-+
(- 192\|2 - 272)cos(x) - 192\|2 - 272
*
sin(x)
+
+-+ 3
(- 144\|2 - 204)cos(x)
+
+-+ 2 +-+
(- 144\|2 - 204)cos(x) + (192\|2 + 272)cos(x)
+
+-+
192\|2 + 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 - 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4
(- 29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 3
(- 87\|2 - 123)\|2 cos(x)
+
+-+ 4+-+ 2
(58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(116\|2 + 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-+ 4+-+ 3
(- 24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ 2
(- 24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+
(96\|2 + 136)\|2 cos(x)
+
+-+ 4+-+
(96\|2 + 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ 2
(72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 - 136)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 - 136)\|2
*
+---------+
| +-+
|2\|2 + 4
|---------
| +-+
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3
(48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 2
(48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2
(- 192\|2 - 272)\|2 cos(x)
+
+-+ 4+-+2
(- 192\|2 - 272)\|2
*
sin(x)
+
+-+ 4+-+2 3
(- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 2
(- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 + 272)\|2 cos(x) + (192\|2 + 272)\|2
/
+-+ 3 +-+ 2
(12\|2 + 17)cos(x) + (12\|2 + 17)cos(x)
+
+-+ +-+
(- 48\|2 - 68)cos(x) - 48\|2 - 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 - 51)cos(x) + (- 36\|2 - 51)cos(x)
+
+-+ +-+
(48\|2 + 68)cos(x) + 48\|2 + 68
+
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 + 4 |2sin(x) - 2
((2\|2 + 2)cos(x) + 2\|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-----------+
+-+4+-+ +-+4+-+ |2sin(x) - 2
(- 2\|2 \|2 cos(x) - 2\|2 \|2 ) |-----------
\| cos(x)
+
-
+---------+
| +-+
4+-+ |2\|2 - 4
4\|2 |---------
| +-+
\|2\|2 - 3
*
atan
+-+ +-+ +-+
((2\|2 + 2)sin(x) + (- 2\|2 - 2)cos(x) - 2\|2 - 2)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ 4|-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-----------+
+-+ 4+-+ +-+ 4+-+ |2sin(x) - 2
((2\|2 + 4)\|2 cos(x) + (2\|2 + 4)\|2 ) |-----------
\| cos(x)
/
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 + 4 |2sin(x) - 2
((\|2 + 2)cos(x) + \|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 + 3
*
ROOT
+-+ 3 +-+ 2
(48\|2 + 68)cos(x) + (48\|2 + 68)cos(x)
+
+-+ +-+
(- 192\|2 - 272)cos(x) - 192\|2 - 272
*
sin(x)
+
+-+ 3
(- 144\|2 - 204)cos(x)
+
+-+ 2 +-+
(- 144\|2 - 204)cos(x) + (192\|2 + 272)cos(x)
+
+-+
192\|2 + 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 - 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4
(- 29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 3
(- 87\|2 - 123)\|2 cos(x)
+
+-+ 4+-+ 2
(58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(116\|2 + 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-+ 4+-+ 3
(24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ 2
(24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 - 136)\|2 cos(x)
+
+-+ 4+-+
(- 96\|2 - 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(- 72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ 2
(- 72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 + 136)\|2 cos(x) + (96\|2 + 136)\|2
*
+---------+
| +-+
|2\|2 + 4
|---------
| +-+
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3
(48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 2
(48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2
(- 192\|2 - 272)\|2 cos(x)
+
+-+ 4+-+2
(- 192\|2 - 272)\|2
*
sin(x)
+
+-+ 4+-+2 3
(- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 2
(- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 + 272)\|2 cos(x) + (192\|2 + 272)\|2
/
+-+ 3 +-+ 2
(12\|2 + 17)cos(x) + (12\|2 + 17)cos(x)
+
+-+ +-+
(- 48\|2 - 68)cos(x) - 48\|2 - 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 - 51)cos(x) + (- 36\|2 - 51)cos(x)
+
+-+ +-+
(48\|2 + 68)cos(x) + 48\|2 + 68
+
+---------+
| +-+ +-----------+
+-+ +-+ |2\|2 + 4 |2sin(x) - 2
((2\|2 + 2)cos(x) + 2\|2 + 2) |--------- |-----------
| +-+ \| cos(x)
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-----------+
+-+4+-+ +-+4+-+ |2sin(x) - 2
(2\|2 \|2 cos(x) + 2\|2 \|2 ) |-----------
\| cos(x)
+
+---------+
| +-+
+-+ 4+-+ |2\|2 - 4
(- \|2 - 1)\|2 |---------
| +-+
\|2\|2 - 3
*
log
+-+ 3 +-+ 2
(48\|2 + 68)cos(x) + (48\|2 + 68)cos(x)
+
+-+ +-+
(- 192\|2 - 272)cos(x) - 192\|2 - 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 - 204)cos(x) + (- 144\|2 - 204)cos(x)
+
+-+ +-+
(192\|2 + 272)cos(x) + 192\|2 + 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 - 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4 +-+ 4+-+ 3
(- 29\|2 - 41)\|2 cos(x) + (- 87\|2 - 123)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(58\|2 + 82)\|2 cos(x) + (116\|2 + 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-+ 4+-+ 3 +-+ 4+-+ 2
(24\|2 + 34)\|2 cos(x) + (24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 - 136)\|2 cos(x) + (- 96\|2 - 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(- 72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(- 72\|2 - 102)\|2 cos(x) + (96\|2 + 136)\|2 cos(x)
+
+-+ 4+-+
(96\|2 + 136)\|2
*
+---------+
| +-+
|2\|2 + 4
|---------
| +-+
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(48\|2 + 68)\|2 cos(x) + (48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 - 272)\|2 cos(x) + (- 192\|2 - 272)\|2
*
sin(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(- 144\|2 - 204)\|2 cos(x) + (- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 + 272)\|2 cos(x) + (192\|2 + 272)\|2
/
+-+ 3 +-+ 2
(12\|2 + 17)cos(x) + (12\|2 + 17)cos(x)
+
+-+ +-+
(- 48\|2 - 68)cos(x) - 48\|2 - 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 - 51)cos(x) + (- 36\|2 - 51)cos(x)
+
+-+ +-+
(48\|2 + 68)cos(x) + 48\|2 + 68
+
+---------+
| +-+
+-+ 4+-+ |2\|2 - 4
(\|2 + 1)\|2 |---------
| +-+
\|2\|2 - 3
*
log
+-+ 3 +-+ 2
(48\|2 + 68)cos(x) + (48\|2 + 68)cos(x)
+
+-+ +-+
(- 192\|2 - 272)cos(x) - 192\|2 - 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 - 204)cos(x) + (- 144\|2 - 204)cos(x)
+
+-+ +-+
(192\|2 + 272)cos(x) + 192\|2 + 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 - 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 - 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4 +-+ 4+-+ 3
(- 29\|2 - 41)\|2 cos(x) + (- 87\|2 - 123)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(58\|2 + 82)\|2 cos(x) + (116\|2 + 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-+ 4+-+ 3
(- 24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ 2
(- 24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 + 136)\|2 cos(x) + (96\|2 + 136)\|2
*
sin(x)
+
+-+ 4+-+ 3 +-+ 4+-+ 2
(72\|2 + 102)\|2 cos(x) + (72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 - 136)\|2 cos(x) + (- 96\|2 - 136)\|2
*
+---------+
| +-+
|2\|2 + 4
|---------
| +-+
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(48\|2 + 68)\|2 cos(x) + (48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 - 272)\|2 cos(x) + (- 192\|2 - 272)\|2
*
sin(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(- 144\|2 - 204)\|2 cos(x) + (- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 + 272)\|2 cos(x) + (192\|2 + 272)\|2
/
+-+ 3 +-+ 2
(12\|2 + 17)cos(x) + (12\|2 + 17)cos(x)
+
+-+ +-+
(- 48\|2 - 68)cos(x) - 48\|2 - 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 - 51)cos(x) + (- 36\|2 - 51)cos(x)
+
+-+ +-+
(48\|2 + 68)cos(x) + 48\|2 + 68
+
+---------+
| +-+
+-+ 4+-+ |2\|2 + 4
(\|2 + 1)\|2 |---------
| +-+
\|2\|2 + 3
*
log
+-+ 3 +-+ 2
(48\|2 - 68)cos(x) + (48\|2 - 68)cos(x)
+
+-+ +-+
(- 192\|2 + 272)cos(x) - 192\|2 + 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 + 204)cos(x) + (- 144\|2 + 204)cos(x)
+
+-+ +-+
(192\|2 - 272)cos(x) + 192\|2 - 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 + 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4 +-+ 4+-+ 3
(- 29\|2 + 41)\|2 cos(x) + (- 87\|2 + 123)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(58\|2 - 82)\|2 cos(x) + (116\|2 - 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-+ 4+-+ 3 +-+ 4+-+ 2
(24\|2 - 34)\|2 cos(x) + (24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 + 136)\|2 cos(x) + (- 96\|2 + 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(- 72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(- 72\|2 + 102)\|2 cos(x) + (96\|2 - 136)\|2 cos(x)
+
+-+ 4+-+
(96\|2 - 136)\|2
*
+---------+
| +-+
|2\|2 - 4
|---------
| +-+
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(48\|2 - 68)\|2 cos(x) + (48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 + 272)\|2 cos(x) + (- 192\|2 + 272)\|2
*
sin(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(- 144\|2 + 204)\|2 cos(x) + (- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 - 272)\|2 cos(x) + (192\|2 - 272)\|2
/
+-+ 3 +-+ 2
(12\|2 - 17)cos(x) + (12\|2 - 17)cos(x)
+
+-+ +-+
(- 48\|2 + 68)cos(x) - 48\|2 + 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 + 51)cos(x) + (- 36\|2 + 51)cos(x)
+
+-+ +-+
(48\|2 - 68)cos(x) + 48\|2 - 68
+
+---------+
| +-+
+-+ 4+-+ |2\|2 + 4
(- \|2 - 1)\|2 |---------
| +-+
\|2\|2 + 3
*
log
+-+ 3 +-+ 2
(48\|2 - 68)cos(x) + (48\|2 - 68)cos(x)
+
+-+ +-+
(- 192\|2 + 272)cos(x) - 192\|2 + 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 + 204)cos(x) + (- 144\|2 + 204)cos(x)
+
+-+ +-+
(192\|2 - 272)cos(x) + 192\|2 - 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 2
(- 58\|2 + 82)\|2 cos(x)
+
+-+ 4+-+
(- 116\|2 + 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4 +-+ 4+-+ 3
(- 29\|2 + 41)\|2 cos(x) + (- 87\|2 + 123)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(58\|2 - 82)\|2 cos(x) + (116\|2 - 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-+ 4+-+ 3
(- 24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ 2
(- 24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 - 136)\|2 cos(x) + (96\|2 - 136)\|2
*
sin(x)
+
+-+ 4+-+ 3 +-+ 4+-+ 2
(72\|2 - 102)\|2 cos(x) + (72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 + 136)\|2 cos(x) + (- 96\|2 + 136)\|2
*
+---------+
| +-+
|2\|2 - 4
|---------
| +-+
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(48\|2 - 68)\|2 cos(x) + (48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 + 272)\|2 cos(x) + (- 192\|2 + 272)\|2
*
sin(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(- 144\|2 + 204)\|2 cos(x) + (- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 - 272)\|2 cos(x) + (192\|2 - 272)\|2
/
+-+ 3 +-+ 2
(12\|2 - 17)cos(x) + (12\|2 - 17)cos(x)
+
+-+ +-+
(- 48\|2 + 68)cos(x) - 48\|2 + 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 + 51)cos(x) + (- 36\|2 + 51)cos(x)
+
+-+ +-+
(48\|2 - 68)cos(x) + 48\|2 - 68
+
+---------+
| +-+
+-+ 4+-+ |2\|2 + 4
(\|2 + 1)\|2 |---------
| +-+
\|2\|2 + 3
*
log
+-+ 3 +-+ 2
(48\|2 - 68)cos(x) + (48\|2 - 68)cos(x)
+
+-+ +-+
(- 192\|2 + 272)cos(x) - 192\|2 + 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 + 204)cos(x) + (- 144\|2 + 204)cos(x)
+
+-+ +-+
(192\|2 - 272)cos(x) + 192\|2 - 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(- 29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(58\|2 - 82)\|2 cos(x) + (116\|2 - 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4 +-+ 4+-+ 3
(29\|2 - 41)\|2 cos(x) + (87\|2 - 123)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(- 58\|2 + 82)\|2 cos(x) + (- 116\|2 + 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-+ 4+-+ 3 +-+ 4+-+ 2
(24\|2 - 34)\|2 cos(x) + (24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 + 136)\|2 cos(x) + (- 96\|2 + 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(- 72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(- 72\|2 + 102)\|2 cos(x) + (96\|2 - 136)\|2 cos(x)
+
+-+ 4+-+
(96\|2 - 136)\|2
*
+---------+
| +-+
|2\|2 - 4
|---------
| +-+
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(48\|2 - 68)\|2 cos(x) + (48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 + 272)\|2 cos(x) + (- 192\|2 + 272)\|2
*
sin(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(- 144\|2 + 204)\|2 cos(x) + (- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 - 272)\|2 cos(x) + (192\|2 - 272)\|2
/
+-+ 3 +-+ 2
(12\|2 - 17)cos(x) + (12\|2 - 17)cos(x)
+
+-+ +-+
(- 48\|2 + 68)cos(x) - 48\|2 + 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 + 51)cos(x) + (- 36\|2 + 51)cos(x)
+
+-+ +-+
(48\|2 - 68)cos(x) + 48\|2 - 68
+
+---------+
| +-+
+-+ 4+-+ |2\|2 + 4
(- \|2 - 1)\|2 |---------
| +-+
\|2\|2 + 3
*
log
+-+ 3 +-+ 2
(48\|2 - 68)cos(x) + (48\|2 - 68)cos(x)
+
+-+ +-+
(- 192\|2 + 272)cos(x) - 192\|2 + 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 + 204)cos(x) + (- 144\|2 + 204)cos(x)
+
+-+ +-+
(192\|2 - 272)cos(x) + 192\|2 - 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(- 29\|2 + 41)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(58\|2 - 82)\|2 cos(x) + (116\|2 - 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4 +-+ 4+-+ 3
(29\|2 - 41)\|2 cos(x) + (87\|2 - 123)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(- 58\|2 + 82)\|2 cos(x) + (- 116\|2 + 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 - 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 - 3
+
+-+ 4+-+ 3
(- 24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ 2
(- 24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 - 136)\|2 cos(x) + (96\|2 - 136)\|2
*
sin(x)
+
+-+ 4+-+ 3 +-+ 4+-+ 2
(72\|2 - 102)\|2 cos(x) + (72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 + 136)\|2 cos(x) + (- 96\|2 + 136)\|2
*
+---------+
| +-+
|2\|2 - 4
|---------
| +-+
\|2\|2 - 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(48\|2 - 68)\|2 cos(x) + (48\|2 - 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 + 272)\|2 cos(x) + (- 192\|2 + 272)\|2
*
sin(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(- 144\|2 + 204)\|2 cos(x) + (- 144\|2 + 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 - 272)\|2 cos(x) + (192\|2 - 272)\|2
/
+-+ 3 +-+ 2
(12\|2 - 17)cos(x) + (12\|2 - 17)cos(x)
+
+-+ +-+
(- 48\|2 + 68)cos(x) - 48\|2 + 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 + 51)cos(x) + (- 36\|2 + 51)cos(x)
+
+-+ +-+
(48\|2 - 68)cos(x) + 48\|2 - 68
+
+---------+
| +-+
+-+ 4+-+ |2\|2 - 4
(- \|2 - 1)\|2 |---------
| +-+
\|2\|2 - 3
*
log
+-+ 3 +-+ 2
(48\|2 + 68)cos(x) + (48\|2 + 68)cos(x)
+
+-+ +-+
(- 192\|2 - 272)cos(x) - 192\|2 - 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 - 204)cos(x) + (- 144\|2 - 204)cos(x)
+
+-+ +-+
(192\|2 + 272)cos(x) + 192\|2 + 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(- 29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(58\|2 + 82)\|2 cos(x) + (116\|2 + 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4 +-+ 4+-+ 3
(29\|2 + 41)\|2 cos(x) + (87\|2 + 123)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(- 58\|2 - 82)\|2 cos(x) + (- 116\|2 - 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-+ 4+-+ 3 +-+ 4+-+ 2
(24\|2 + 34)\|2 cos(x) + (24\|2 + 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 - 136)\|2 cos(x) + (- 96\|2 - 136)\|2
*
sin(x)
+
+-+ 4+-+ 3
(- 72\|2 - 102)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(- 72\|2 - 102)\|2 cos(x) + (96\|2 + 136)\|2 cos(x)
+
+-+ 4+-+
(96\|2 + 136)\|2
*
+---------+
| +-+
|2\|2 + 4
|---------
| +-+
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(48\|2 + 68)\|2 cos(x) + (48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 - 272)\|2 cos(x) + (- 192\|2 - 272)\|2
*
sin(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(- 144\|2 - 204)\|2 cos(x) + (- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 + 272)\|2 cos(x) + (192\|2 + 272)\|2
/
+-+ 3 +-+ 2
(12\|2 + 17)cos(x) + (12\|2 + 17)cos(x)
+
+-+ +-+
(- 48\|2 - 68)cos(x) - 48\|2 - 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 - 51)cos(x) + (- 36\|2 - 51)cos(x)
+
+-+ +-+
(48\|2 + 68)cos(x) + 48\|2 + 68
+
+---------+
| +-+
+-+ 4+-+ |2\|2 - 4
(\|2 + 1)\|2 |---------
| +-+
\|2\|2 - 3
*
log
+-+ 3 +-+ 2
(48\|2 + 68)cos(x) + (48\|2 + 68)cos(x)
+
+-+ +-+
(- 192\|2 - 272)cos(x) - 192\|2 - 272
*
sin(x)
+
+-+ 3 +-+ 2
(- 144\|2 - 204)cos(x) + (- 144\|2 - 204)cos(x)
+
+-+ +-+
(192\|2 + 272)cos(x) + 192\|2 + 272
*
+-----------+2
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+ 3
(- 29\|2 - 41)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(58\|2 + 82)\|2 cos(x) + (116\|2 + 164)\|2 cos(x)
*
sin(x)
+
+-+ 4+-+ 4 +-+ 4+-+ 3
(29\|2 + 41)\|2 cos(x) + (87\|2 + 123)\|2 cos(x)
+
+-+ 4+-+ 2 +-+ 4+-+
(- 58\|2 - 82)\|2 cos(x) + (- 116\|2 - 164)\|2 cos(x)
*
+---------+
| +-+ +------------+ +-----------+
|2\|2 + 4 |- cos(x) - 1 |2sin(x) - 2
|--------- |------------ |-----------
| +-+ \| cos(x) \| cos(x)
\|2\|2 + 3
+
+-+ 4+-+ 3
(- 24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ 2
(- 24\|2 - 34)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(96\|2 + 136)\|2 cos(x) + (96\|2 + 136)\|2
*
sin(x)
+
+-+ 4+-+ 3 +-+ 4+-+ 2
(72\|2 + 102)\|2 cos(x) + (72\|2 + 102)\|2 cos(x)
+
+-+ 4+-+ +-+ 4+-+
(- 96\|2 - 136)\|2 cos(x) + (- 96\|2 - 136)\|2
*
+---------+
| +-+
|2\|2 + 4
|---------
| +-+
\|2\|2 + 3
*
+-----------+
|2sin(x) - 2
4|-----------
\| cos(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(48\|2 + 68)\|2 cos(x) + (48\|2 + 68)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(- 192\|2 - 272)\|2 cos(x) + (- 192\|2 - 272)\|2
*
sin(x)
+
+-+ 4+-+2 3 +-+ 4+-+2 2
(- 144\|2 - 204)\|2 cos(x) + (- 144\|2 - 204)\|2 cos(x)
+
+-+ 4+-+2 +-+ 4+-+2
(192\|2 + 272)\|2 cos(x) + (192\|2 + 272)\|2
/
+-+ 3 +-+ 2
(12\|2 + 17)cos(x) + (12\|2 + 17)cos(x)
+
+-+ +-+
(- 48\|2 - 68)cos(x) - 48\|2 - 68
*
sin(x)
+
+-+ 3 +-+ 2
(- 36\|2 - 51)cos(x) + (- 36\|2 - 51)cos(x)
+
+-+ +-+
(48\|2 + 68)cos(x) + 48\|2 + 68
/
+---------+ +---------+
| +-+ | +-+
+-+ |2\|2 - 4 |2\|2 + 4
(2\|2 + 2) |--------- |---------
| +-+ | +-+
\|2\|2 - 3 \|2\|2 + 3


--
Waldek Hebisch
heb...@math.uni.wroc.pl

[clicl...@freenet.de]

By now a number of messages were sent to this thread which did not make
it to the Aioe.org NNTP server in Italy.

on May 23, 04:39 PM by Albert D. Rich
on May 23, 08:01 PM by Waldek Hebisch
on May 24, 10:01 AM by Andreas Dieckmann

The timestamps are those displayed on Drexel's Math Forum. As the last
message is also missing on Google Groups, it was probably posted on Math
Forum only. Hopefully communication will be restored soon!

-

This is my antiderivative for problem #45 from Charlwood's appendix:

SQRT(2)*SQRT(SEC(x)+1)*SQRT(SEC(x)-1)*COT(x)*(SQRT(SQRT(2)-1)*AT~
AN(SQRT(SQRT(2)-1)*(SQRT(SEC(x)+1)-SQRT(SEC(x)-1)-SQRT(2))/(SQRT~
(2)*SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))))-SQRT(SQRT(2)+1)*ATAN(S~
QRT(SQRT(2)+1)*(SQRT(SEC(x)+1)-SQRT(SEC(x)-1)-SQRT(2))/(SQRT(2)*~
SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))))+SQRT(SQRT(2)-1)*ATANH(SQRT~
(2*SQRT(2)+2)*SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))/(SQRT(SEC(x)+1~
)-SQRT(SEC(x)-1)+SQRT(2)))-SQRT(SQRT(2)+1)*ATANH(SQRT(2*SQRT(2)-~
2)*SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))/(SQRT(SEC(x)+1)-SQRT(SEC(~
x)-1)+SQRT(2))))

which holds on the entire complex plane.

Martin.

[clicl...@freenet.de]

clicl...@freenet.de schrieb:

>
> This is my antiderivative for problem #45 from Charlwood's appendix:
>
> SQRT(2)*SQRT(SEC(x)+1)*SQRT(SEC(x)-1)*COT(x)*(SQRT(SQRT(2)-1)*AT~
> AN(SQRT(SQRT(2)-1)*(SQRT(SEC(x)+1)-SQRT(SEC(x)-1)-SQRT(2))/(SQRT~
> (2)*SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))))-SQRT(SQRT(2)+1)*ATAN(S~
> QRT(SQRT(2)+1)*(SQRT(SEC(x)+1)-SQRT(SEC(x)-1)-SQRT(2))/(SQRT(2)*~
> SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))))+SQRT(SQRT(2)-1)*ATANH(SQRT~
> (2*SQRT(2)+2)*SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))/(SQRT(SEC(x)+1~
> )-SQRT(SEC(x)-1)+SQRT(2)))-SQRT(SQRT(2)+1)*ATANH(SQRT(2*SQRT(2)-~
> 2)*SQRT(SQRT(SEC(x)+1)-SQRT(SEC(x)-1))/(SQRT(SEC(x)+1)-SQRT(SEC(~
> x)-1)+SQRT(2))))
>
> which holds on the entire complex plane.
>

And here are versions of the antiderivatives for problems #12 and #13
from Charlwood's appendix that do not involve the imaginary unit:

x*ATAN(SQRT(1 - x^2) + x) - 1/2*ASIN(x)
- SQRT(3)/4*ATAN((2*x^2 - 1)/SQRT(3))
+ SQRT(3)/4*ATAN((SQRT(3)*x - 1)/SQRT(1 - x^2))
+ SQRT(3)/4*ATAN((SQRT(3)*x + 1)/SQRT(1 - x^2))
- 1/8*LN(x^4 - x^2 + 1) - 1/4*ATANH(x*SQRT(1 - x^2))

- SQRT(1 - x^2)*ATAN(SQRT(1 - x^2) + x) - 1/2*ASIN(x)
- SQRT(3)/4*ATAN((2*x^2 - 1)/SQRT(3))
+ SQRT(3)/4*ATAN((SQRT(3)*x - 1)/SQRT(1 - x^2))
+ SQRT(3)/4*ATAN((SQRT(3)*x + 1)/SQRT(1 - x^2))
+ 1/8*LN(x^4 - x^2 + 1) + 1/4*ATANH(x*SQRT(1 - x^2))

While the discontinuity of Albert's antiderivatives at x = -1 and x = +1
is not serious, the above are more compact and also fully continuous.

The elliptic integral #49 in the appendix should be taken for a misprint
in view of Charlwood's statement that he considers "the ability of three
computer algebra systems (CAS) to evaluate [integrals] in closed-form,
appealing only to the class of real, elementary functions", so I see no
reason to remove the imaginary unit here. It can be done though.

The other antiderivatives in Albert's file are real already.

Martin.

[Albert Rich]

I just posted a revised pdf file of the Charlwood Fifty integration test-suite at

http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf

that shows improved antiderivatives for problems #12, #13 (that Martin suggested) and #41. Please disregard the previous version of the file. Also posted is a Mathematica package file of the test-suite in machine readable form at

http://www.apmaths.uwo.ca/~arich/CharlwoodProblems.m

Not having heard back from Professor Charlwood, I took the liberty of changing the integrand of problem #49 from arcsin(x*sqrt(1-x^2)) to arcsin(x/sqrt(1-x^2)). This was done so all the integrands and antiderivatives in the test-suite would involve only elementary functions and operators.

Hopefully all the antiderivatives in Charlwood Fifty test-suite are now optimal...

Albert

[clicl...@freenet.de]

Albert Rich schrieb:

It would have been nice if Prof. Charlwood could have thrown light on
problem #49 from his appendix: Did he really want his students to work
on an elementary evaluation of INT(ASIN(x*SQRT(1-x^2)), x) and fail?
Anyway, here is a real version of the elliptic result:

x*ASIN(x*SQRT(1-x^2))+2*SQRT(1-x^2)*SQRT(x^4-x^2+1)/(2-x^2)+SQRT~
(x^4/(2-x^2)^2)*SQRT(x^4-x^2+1)/(2*x^2*SQRT((x^4-x^2+1)/(2-x^2)^~
2))*(EL_F(ASIN(2*SQRT(1-x^2)/(2-x^2)),SQRT(3)/2)-4*EL_E(ASIN(2*S~
QRT(1-x^2)/(2-x^2)),SQRT(3)/2))

where the incomplete elliptic integrals are defined as:

EL_F(phi, k) := INT(1/SQRT(1 - k^2*SIN(t)^2), t, 0, phi)

EL_E(phi, k) := INT(SQRT(1 - k^2*SIN(t)^2), t, 0, phi)

Martin.

[Nasser M. Abbasi]

On 6/6/2013 11:23 AM, clicl...@freenet.de wrote:
>

>
> It would have been nice if Prof. Charlwood could have thrown light on
> problem #49 from his appendix: Did he really want his students to work
> on an elementary evaluation of INT(ASIN(x*SQRT(1-x^2)), x) and fail?

fyi,

In http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf

#49 is written as INT(ASIN(x/SQRT(1-x^2)), x)

I just checked the Charlwood’s 2008 paper, and it should be as
you have shown it (i.e. multiplication not division), so there is a
typo in the above pdf file.

> Anyway, here is a real version of the elliptic result:
>
> x*ASIN(x*SQRT(1-x^2))+2*SQRT(1-x^2)*SQRT(x^4-x^2+1)/(2-x^2)+SQRT~
> (x^4/(2-x^2)^2)*SQRT(x^4-x^2+1)/(2*x^2*SQRT((x^4-x^2+1)/(2-x^2)^~
> 2))*(EL_F(ASIN(2*SQRT(1-x^2)/(2-x^2)),SQRT(3)/2)-4*EL_E(ASIN(2*S~
> QRT(1-x^2)/(2-x^2)),SQRT(3)/2))
>
> where the incomplete elliptic integrals are defined as:
>
> EL_F(phi, k) := INT(1/SQRT(1 - k^2*SIN(t)^2), t, 0, phi)
>
> EL_E(phi, k) := INT(SQRT(1 - k^2*SIN(t)^2), t, 0, phi)
>
> Martin.
>

btw, this is what Mathematica gives for this one:

In[1]:= Integrate[ArcSin[x*Sqrt[1 - x^2]], x]

Out[1]= x ArcSin[x Sqrt[1 - x^2]] + (1/Sqrt[
1 - x^2 + x^4])(1 - x^2)^(
3/2) (2 + 2/(-1 + x^2)^2 + 2/(-1 + x^2) + (
2 (-1)^(5/6) Sqrt[(-1 + (-1)^(1/3) + x^2)/(-1 + x^2)] Sqrt[
1 - (-1)^(2/3)/(-1 + x^2)]
EllipticE[I ArcSinh[(-1)^(1/3)/Sqrt[1 - x^2]], (-1)^(2/3)])/
Sqrt[1 - x^2] - ((-1)^(2/3) Sqrt[3 + (3 (-1)^(1/3))/(-1 + x^2)]
Sqrt[1 - (-1)^(2/3)/(-1 + x^2)]
EllipticF[I ArcSinh[(-1)^(1/3)/Sqrt[1 - x^2]], (-1)^(2/3)])/
Sqrt[1 - x^2])

Maple 17 could not seem to be able to do it. returned unevaluated.

--Nasser

[clicl...@freenet.de]

"Nasser M. Abbasi" schrieb:

>
> On 6/6/2013 11:23 AM, clicl...@freenet.de wrote:
> >
> > It would have been nice if Prof. Charlwood could have thrown light on
> > problem #49 from his appendix: Did he really want his students to work
> > on an elementary evaluation of INT(ASIN(x*SQRT(1-x^2)), x) and fail?
> >
>

> In http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf
>
> #49 is written as INT(ASIN(x/SQRT(1-x^2)), x)
>
> I just checked the Charlwood�s 2008 paper, and it should be as
> you have shown it (i.e. multiplication not division), so there is a
> typo in the above pdf file.
>

Oh, as Albert Rich wrote, he "took the liberty of changing the integrand

of problem #49 from arcsin(x*sqrt(1-x^2)) to arcsin(x/sqrt(1-x^2)). This
was done so all the integrands and antiderivatives in the test-suite
would involve only elementary functions and operators."

>

> [...]

>
> Maple 17 could not seem to be able to do it. returned unevaluated.
>

Strange that Maple can't do this one. Derive 6.10 immediately converts
the integral to

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

and Maple should be able to cope with the algebraic integral that
remains (the Derive integrator has no knowledge of the canonical
elliptic integrals).

Martin.

[clicl...@freenet.de]

Albert Rich schrieb:

>
> I just posted a revised pdf file of the Charlwood Fifty integration
> test-suite at
>
> http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf
>

> [...] Also posted is a Mathematica package file of the test-suite in

> machine readable form at
>
> http://www.apmaths.uwo.ca/~arich/CharlwoodProblems.m
>

> [...] Hopefully all the antiderivatives in Charlwood Fifty test-suite
> are now optimal...
>

I haven't checked the evaluations in the file systematically, but a new
look has revealed further possibilities for improvement.

The present solutions of problems #21 and #22 from Charlwood's appendix,
INT(x^3*ASIN(x)/SQRT(1-x^4), x) and INT(x^3*ASEC(x)/SQRT(x^4-1), x), can
be written as:

1/4*(x*SQRT(1-x^4)/SQRT(1-x^2)
+ LN(1-x^2) - LN(-x + x^3 + SQRT(1-x^2)*SQRT(1-x^4)))
- 1/2*SQRT(1-x^4)*ASIN(x)

1/2*(SQRT(x^4-1)*ASEC(x) - SQRT(x^4-1)/(x*SQRT(1 - 1/x^2))
- LN(x - x^3) + LN(1 - x^2 - x*SQRT(x^4-1)*SQRT(1 - 1/x^2)))

In my eyes, ATANH constitutes a more natural option here than LN:

1/4*(x*SQRT(1-x^4)/SQRT(1-x^2) + ATANH(x*SQRT(1-x^2)/SQRT(1-x^4)))
- 1/2*SQRT(1-x^4)*ASIN(x)

1/2*(SQRT(x^4-1)*ASEC(x) - SQRT(x^4-1)/(x*SQRT(1 - 1/x^2))
+ ATANH(x*SQRT(1 - 1/x^2)/SQRT(x^4-1)))

Or, using piecewise-constant prefactors:

SQRT(1-x^4)/(4*SQRT(1-x^2)*SQRT(1+x^2))
*(x*SQRT(1+x^2) + ATANH(x/SQRT(1+x^2))) - 1/2*SQRT(1-x^4)*ASIN(x)

1/2*(SQRT(x^4-1)*ASEC(x) - x*SQRT(x^2+1)/SQRT(x^4-1)*SQRT(1 - 1/x^2)
*(SQRT(x^2+1) - ATANH(1/SQRT(x^2+1))))

A much simpler evaluation of integral #48 from the appendix is:

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

Finally, in the antiderivative #50, the rationalization of radicand
denominators in the numerical prefactors may be considered:

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

Martin.

[Albert Rich]

On Saturday, June 8, 2013 7:17:17 AM UTC-10, clicl...@freenet.de wrote:

> I haven't checked the evaluations in the file systematically, but a new
> look has revealed further possibilities for improvement.

> ...

I just posted a revised Charlwood Fifty test-suite at

http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf

that includes the antiderivatives for problems #21, #22, #48 and #50 suggested by Martin. Please disregard the 7 previous iterations of this file that have been posted. Presumably only a finite number of improvements are possible...

Albert

[clicl...@freenet.de]

Albert Rich schrieb:
>
> [...] Presumably only a finite number of improvements are possible...
>

Are you sure? Here are shorter versions of the antiderivatives #41, #42
and #44 from Charlwood's appendix:

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

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

INT(SIN(x)/SQRT(1 - SIN(x)^6), x) =
SQRT(3)/6*ATANH(SQRT(3)*COS(x)*(1 + SIN(x)^2)/(2*SQRT(1 - SIN(x)^6)))

These were arrived at by piecewise-constant acrobatics.

Martin.

[clicl...@freenet.de]

clicl...@freenet.de schrieb:

>
> Albert Rich schrieb:
> >
> > [...] Presumably only a finite number of improvements are
> > possible...
>

> Are you sure? [...]

For problem #41 from Charlwood's appendix, further piecewise-constant
acrobatics results in:

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

- 4*ATANH(COS(x)/SQRT(1 + SIN(x)))

Martin.

[Albert Rich]

On Saturday, June 8, 2013 11:12:56 AM UTC-10, clicl...@freenet.de wrote:

> Albert Rich schrieb:
>
>> [...] Presumably only a finite number of improvements are possible...
>

> Are you sure? [...]

You are severely testing my cherished belief that optimality exists... :=)

I just posted a revised Charlwood Fifty test-suite at

http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf

that includes your most recent antiderivatives for problems #41, #42 and #44. Please disregard the 8 previous versions.

Albert

[Nasser M. Abbasi]

On 6/9/2013 4:58 AM, Albert Rich wrote:

fyi;

This is another hard integral posted at the Mathematica newsgroup
last night.

I do not know if you like to add it to your test suite as Rubi4 and
Mathematica 9.01 have some hard time with it.


ClearAll["`*"];
in = -((-1 + Cos[z])/(z^2 (r^2 + z^2)^2 (z^2 - 4 \[Pi]^2)^2));
Assuming[r > 0, Int[in, {z, -Infinity, Infinity}]]

Not evaluated. Maple 17 does it fast:

restart;
integrand:=-((-1 + cos(z))/(z^2*(r^2 + z^2)^2 *(z^2 - 4*Pi^2)^2));
int(integrand, z=-infinity..infinity) assuming r>0;

-(1/32)*(64*Pi^6*sinh(r)*r-64*Pi^6*cosh(r)*r+16*Pi^4*sinh(r)*
r^3-16*Pi^4*cosh(r)*r^3+192*Pi^6*sinh(r)-192*Pi^6*cosh(r)-
128*Pi^6*r+112*Pi^4*sinh(r)*r^2-112*Pi^4*cosh(r)*r^2-96*Pi^4*
r^3-28*Pi^2*r^5-3*r^7+192*Pi^6+112*Pi^4*r^2)/(r^5*(4*Pi^2+r^2)^3*Pi^3)

Mathematica takes about 2-3 minutes and gives this:

ClearAll["`*"];
in = -((-1 + Cos[z])/(z^2 (r^2 + z^2)^2 (z^2 - 4 Pi^2)^2));
Assuming[r > 0, Integrate[in, {z, -Infinity, Infinity}]]


-(1/(128 \[Pi]^4 r^5 (4 \[Pi]^2 + r^2)^3))(768 \[Pi]^7 +
448 \[Pi]^5 r^2 - 92 \[Pi]^3 r^5 - 11 \[Pi] r^7 +
40 I \[Pi]^2 r^5 CosIntegral[2 \[Pi]] +
2 I r^7 CosIntegral[2 \[Pi]] -
40 I \[Pi]^2 r^5 ExpIntegralEi[-2 I \[Pi]] -
2 I r^7 ExpIntegralEi[-2 I \[Pi]] +
16 \[Pi]^(7/2)
r (4 \[Pi]^2 +
5 r^2) MeijerG[{{1/2, 1}, {}}, {{-(1/2), 1/2, 1}, {0}}, -((I r)/
2), 1/2] +
16 \[Pi]^(7/2)
r (4 \[Pi]^2 +
5 r^2) MeijerG[{{1/2, 1}, {}}, {{-(1/2), 1/2, 1}, {0}}, (I r)/2,
1/2] + 128 \[Pi]^(11/2)
r MeijerG[{{1/2, 1}, {}}, {{-(1/2), 1, 3/2}, {0}}, -((I r)/2), 1/
2] + 32 \[Pi]^(7/2)
r^3 MeijerG[{{1/2, 1}, {}}, {{-(1/2), 1, 3/2}, {0}}, -((I r)/2),
1/2] + 128 \[Pi]^(11/2)
r MeijerG[{{1/2, 1}, {}}, {{-(1/2), 1, 3/2}, {0}}, (I r)/2, 1/
2] + 32 \[Pi]^(7/2)
r^3 MeijerG[{{1/2, 1}, {}}, {{-(1/2), 1, 3/2}, {0}}, (I r)/2, 1/
2] + 40 \[Pi]^2 r^5 SinIntegral[2 \[Pi]] +
2 r^7 SinIntegral[2 \[Pi]])

--Nasser

[Richard Fateman]

On 6/8/2013 10:17 AM, clicl...@freenet.de wrote:
> I haven't checked the evaluations in the file systematically, but a new
> look has revealed further possibilities for improvement.
>
> The present solutions of problems #21 and #22 from Charlwood's appendix,
> INT(x^3*ASIN(x)/SQRT(1-x^4), x) and INT(x^3*ASEC(x)/SQRT(x^4-1), x), can
> be written as:
>
> 1/4*(x*SQRT(1-x^4)/SQRT(1-x^2)
> + LN(1-x^2) - LN(-x + x^3 + SQRT(1-x^2)*SQRT(1-x^4)))
> - 1/2*SQRT(1-x^4)*ASIN(x)

How about
((asinh(x))/4) - ((sqrt(1 - x^4) * asin(x))/2) + ((x * sqrt(x^2 + 1))/4)

from Macsyma. Which has the advantage of using no logs, just asin() and
asinh(). Since the integrand has asin, I think it is nice.

I have tried only a few others in the file, and Macsyma stumbled on one
or two of them. I am not inclined generally to spend my time to see if
the Macsyma solutions are smaller, neater, continuous etc, and maybe
it is of no interest since Macsyma is not easily available. But then
neither is Derive.. I suppose I can run the file through Macsyma if
someone else is willing to look at the results :)

RJF

[Richard Fateman]

On 6/8/2013 10:17 AM, clicl...@freenet.de wrote:

> A smaller representation of the solution to #22 from Charlwood's appendix,...

INT(x^3*ASEC(x)/SQRT(x^4-1), x)

is

((a * (sqrt(x^2 - 1) * asec(x) - 1))/2) + ((log(((a + 1)/(a - 1))))/4)

with a=sqrt(x^2+1).


I have not investigated how this fairs with respect to maximal
continuity.

RJF

[Nasser M. Abbasi]

On 6/9/2013 4:58 AM, Albert Rich wrote:

fyi;

I've update the listing of the first 10 integrals. Added your optimal
results there, and also added result from Sage 5.4.

http://www.12000.org/my_notes/ten_hard_integrals/index.htm

Each integral is now on a separate web page instead of them
all on the same page to make it easier to see the results
of each.

There is also a pdf file.

Here is a quick summary of the first 10 integral results,
I just counted if CAS gave result or not. No checking for
anything else. Some results are clearly not optimal and
few results take many pages.

1. Mathematica: did 1,2,3,4,6,8,9,10 did not: 5,7
2. Maple: did 1,2,3,4,5,6,7,8 did not: 9,10
3. Rubi4: did 2,3(?),4,5,6,7,8,9,10 did not: 1
4. Sage: did: 1,2,3,4,7,9 did not: 5,6,8,10

I do not understand Rubi4 result for 3. I do not know what
Subst[....] is supposed to mean, but I have not looked it
up, I am sure it is explained somewhere. Any errors, please
let me know.

--Nasser

[Albert Rich]

On Sunday, June 9, 2013 3:37:09 PM UTC-10, Richard Fateman wrote:

> INT(x^3*ASEC(x)/SQRT(x^4-1), x)
>
> is
>
> ((a * (sqrt(x^2 - 1) * asec(x) - 1))/2) + ((log(((a + 1)/(a - 1))))/4)
>
> with a=sqrt(x^2+1).
>
> I have not investigated how this fairs with respect to maximal
> continuity.

Unfortunately, that is not a valid antiderivative since subtracting its derivative from the original integrand and substituting -2 for x does not equal zero.

Albert

[Albert Rich]

On Sunday, June 9, 2013 3:12:08 PM UTC-10, Richard Fateman wrote:

> (For Charlwood problem #21) How about

>
> ((asinh(x))/4) - ((sqrt(1 - x^4) * asin(x))/2) + ((x * sqrt(x^2 + 1))/4)
>
> from Macsyma. Which has the advantage of using no logs, just asin() and
> asinh(). Since the integrand has asin, I think it is nice.

Very nice. I just posted a revised Charlwood Fifty test-suite at

http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf

Please disregard previous versions.

Albert

[Albert Rich]

On Sunday, June 9, 2013 4:25:45 AM UTC-10, Nasser M. Abbasi wrote:

> This is another hard integral posted at the Mathematica newsgroup
> last night.
>
> I do not know if you like to add it to your test suite as Rubi4 and
> Mathematica 9.01 have some hard time with it.
>
> ClearAll["`*"];
>
> in = -((-1 + Cos[z])/(z^2 (r^2 + z^2)^2 (z^2 - 4 \[Pi]^2)^2));
>
> Assuming[r > 0, Int[in, {z, -Infinity, Infinity}]]

Currently Rubi only has rules for finding indefinite integrals, and the test-suite only has indefinite integration problems.

Albert

[clicl...@freenet.de]

"Nasser M. Abbasi" schrieb:

>
> I've update the listing of the first 10 integrals. Added your optimal
> results there, and also added result from Sage 5.4.
>
> http://www.12000.org/my_notes/ten_hard_integrals/index.htm
>
> Each integral is now on a separate web page instead of them
> all on the same page to make it easier to see the results
> of each.
>
> There is also a pdf file.
>
> Here is a quick summary of the first 10 integral results,
> I just counted if CAS gave result or not. No checking for
> anything else. Some results are clearly not optimal and
> few results take many pages.
>
> 1. Mathematica: did 1,2,3,4,6,8,9,10 did not: 5,7
> 2. Maple: did 1,2,3,4,5,6,7,8 did not: 9,10
> 3. Rubi4: did 2,3(?),4,5,6,7,8,9,10 did not: 1
> 4. Sage: did: 1,2,3,4,7,9 did not: 5,6,8,10
>
> I do not understand Rubi4 result for 3. I do not know what
> Subst[....] is supposed to mean, but I have not looked it
> up, I am sure it is explained somewhere. Any errors, please
> let me know.
>

I noticed that your MMA and Rubi integrands for problem #7 have and
extra outer root.

I strongly suspect that "Sage" always means "Maxima" (not e.g. "Sympy"):
see the Runtime Error Message for problem #8.

Rubi's SUBST means "substitute": SUBST(a+b,b,c+d) = a+(c+d). Since
Rubi's answer to problem #3 contains an unresolved integral inside the
SUBST statemant, it must be counted as "not done".

Martin.

[Nasser M. Abbasi]

On 6/10/2013 5:23 AM, clicl...@freenet.de wrote:

>
> I noticed that your MMA and Rubi integrands for problem #7 have and
> extra outer root.

opps, thanks. I just corrected this

http://www.12000.org/my_notes/ten_hard_integrals/index.htm

>
> I strongly suspect that "Sage" always means "Maxima" (not e.g. "Sympy"):
> see the Runtime Error Message for problem #8.
>
> Rubi's SUBST means "substitute": SUBST(a+b,b,c+d) = a+(c+d). Since
> Rubi's answer to problem #3 contains an unresolved integral inside the
> SUBST statemant, it must be counted as "not done".
>
> Martin.
>

Ok, the score now looks like this then: (Mathematica gained one, since
it can now do 7, and Rubi4 lost one, since 3 did not count as did)

1. Mathematica: did 1,2,3,4,6,7,8,9,10 did not: 5

2. Maple: did 1,2,3,4,5,6,7,8 did not: 9,10

3. Rubi4: did 2,4,5,6,7,8,9,10 did not: 1,3

4. Sage: did: 1,2,3,4,7,9 did not: 5,6,8,10

btw, Sage 5.4 gives

-1/3 ln(cos(x)^3+1)

for #7 while the optimal result is given as

-2/3 arctanh(sqrt(1+sec(x)^3))

Is the reason the optimal form is prefered over Sage's becuase
it does not contain a log even though Sage answer is smaller in
terms of leaf count?

ps. I used Sage notebook web server http://www.sagenb.org/ versionn
5.4

--Nasser

[Nasser M. Abbasi]

On 6/10/2013 6:14 AM, Nasser M. Abbasi wrote:

> btw, Sage 5.4 gives
>
> -1/3 ln(cos(x)^3+1)
>
> for #7 while the optimal result is given as
>
> -2/3 arctanh(sqrt(1+sec(x)^3))
>

Opps, please ignore the above. Just noticed after hitting the
send button that I did not put a sqrt() for #7 for sage input.

Corrected it again. So I had 2 errors in 7 :(

thanks,
--Nasser

[Richard Fateman]

It checks out in Macsyma. In separate correspondence with Albert Rich,
he suggested looking at how Macsyma defines the derivative of arcsecant.
Indeed, Macsyma's definition differs by a sign for negative argument.

For Macsyma's definition, the integral is correct. For Maple or
Mathematica or Maxima, the integral needs an extra abs().

I suppose the dlmf would provide the definitive form, but it seems to
be offline at the moment. I'm guessing it agrees with Maxima, not Macsyma.

RJF

[clicl...@freenet.de]

Richard Fateman schrieb:
>
> [...]

>
> I suppose the dlmf would provide the definitive form, but it seems to
> be offline at the moment. I'm guessing it agrees with Maxima, not
> Macsyma.
>

DLMF had been off-line today but is on-line again now.

Martin.

[Waldek Hebisch]

Richard Fateman <fat...@cs.berkeley.edu> wrote:
> On 6/9/2013 7:40 PM, Albert Rich wrote:
> > On Sunday, June 9, 2013 3:37:09 PM UTC-10, Richard Fateman wrote:
> >
> >> INT(x^3*ASEC(x)/SQRT(x^4-1), x)
> >>
> >> is
> >>
> >> ((a * (sqrt(x^2 - 1) * asec(x) - 1))/2) + ((log(((a + 1)/(a -
> >> 1))))/4)
> >>
> >> with a=sqrt(x^2+1).
> >>
> >> I have not investigated how this fairs with respect to maximal
> >> continuity.
> >
> > Unfortunately, that is not a valid antiderivative since subtracting
> > its derivative from the original integrand and substituting -2 for x
> > does not equal zero.
> >
> > Albert
> >
>
> It checks out in Macsyma. In separate correspondence with Albert Rich,
> he suggested looking at how Macsyma defines the derivative of arcsecant.
> Indeed, Macsyma's definition differs by a sign for negative argument.
>
> For Macsyma's definition, the integral is correct. For Maple or
> Mathematica or Maxima, the integral needs an extra abs().
>

FYI in FriCAS:

(19) -> D(asec(x), x)

1
(19) ----------
+------+
| 2
x\|x - 1
Type: Expression(Integer)

Numerical evaluation of this at negative numbers gives wrong
value because standard branch choice for sqrt is wrong in
this case: in the formula above taking positive branch of
sqrt for x > 1 implies negative branch for x < -1. Using

1/(x^2*sqrt(1 - 1/x^2))

as formula for derivative avoids this problem. OTOH
previous formula have its own advantages, so the choice
is not so clear.

Concerning Macsyma result: for "branch correct" result
Macsyma should use a = sqrt(x^4 - 1)/sqrt(x^2-1).

--
Waldek Hebisch
heb...@math.uni.wroc.pl

[Nasser M. Abbasi]

On 6/10/2013 9:44 PM, Waldek Hebisch wrote:

>
> FYI in FriCAS:
>


Fyi, I also added FriCAS results to the table. I used 1.2, build it
from the source tar file on linux, used sbcl lisp installation.

It was fast, even on a VBox. First time I used it. So I hope I
did not make any errors. Updated the page :

http://www.12000.org/my_notes/ten_hard_integrals/index.htm

The score now stands like this:

1. Mathematica: did 1,2,3,4,6,7,8,9,10 did not: 5
2. Maple: did 1,2,3,4,5,6,7,8 did not: 9,10
3. Rubi4: did 2,4,5,6,7,8,9,10 did not: 1,3
4. Sage: did: 1,2,3,4,7,9 did not: 5,6,8,10

5. Fricas: did 1,2,3,4,5,6,7,10 did not: 5,8,9

If you spot any error, please let me know and will correct it.

--Nasser

[clicl...@freenet.de]

"Nasser M. Abbasi" schrieb:

>
> Fyi, I also added FriCAS results to the table. I used 1.2, build it
> from the source tar file on linux, used sbcl lisp installation.
>
> It was fast, even on a VBox. First time I used it. So I hope I
> did not make any errors. Updated the page :
>
> http://www.12000.org/my_notes/ten_hard_integrals/index.htm
>
> The score now stands like this:
>
> 1. Mathematica: did 1,2,3,4,6,7,8,9,10 did not: 5
> 2. Maple: did 1,2,3,4,5,6,7,8 did not: 9,10
> 3. Rubi4: did 2,4,5,6,7,8,9,10 did not: 1,3
> 4. Sage: did: 1,2,3,4,7,9 did not: 5,6,8,10
> 5. Fricas: did 1,2,3,4,5,6,7,10 did not: 5,8,9
>
> If you spot any error, please let me know and will correct it.
>

For FriCAS you have listed problem #5 as both "did" and "did not".

It would be informative if you could signal non-elementary answers by
putting the corresponding problem numbers in parentheses, say.

Martin.

[Nasser M. Abbasi]

On 6/11/2013 6:37 AM, clicl...@freenet.de wrote:

>
> For FriCAS you have listed problem #5 as both "did" and "did not".
>

opps, sorry, typo. Fricas "did not" do 5.

> It would be informative if you could signal non-elementary answers by
> putting the corresponding problem numbers in parentheses, say.

Sure. A quick look shows that of those solved, #5 by Rubi
and Maple had non-elementary anti-derivative (Elliptic function) in
them. This is the one Mathematica, Sage and FriCAS did not
give an answer for.

In #6,#7,#8,#9 Mathematica gave non-elementary anti-derivative (also using
Elliptic functions).

If I overlooked something, please feel free to correct, all
other anti-derivative are using elementary function as far as I
see (exp, log, radicals, trig and inverse trigs and +,-,/,*)

Here is an updated score table using your proposed notation:

1. Mathematica: did 1,2,3,4,(6),(7),(8),(9),10 did not: 5
2. Maple: did 1,2,3,4,(5),6,7,8 did not: 9,10
3. Rubi4: did 2,4,(5),6,7,8,9,10 did not: 1,3
4. Sage: did: 1,2,3,7,9 did not: 4,5,6,8,10
5. Fricas: did 1,2,3,4,6,7,10 did not: 5,8,9

ps. I moved Sage's 4 from the "did" to the "did not" since the
answer still contained another integral in it, so to be
fair to other CAS's it should have been counted as "did not".
sorry Sage :(

thanks,
--Nasser

[Nasser M. Abbasi]

On 6/10/2013 5:23 AM, clicl...@freenet.de wrote:

>

> I strongly suspect that "Sage" always means "Maxima" (not e.g. "Sympy"):
> see the Runtime Error Message for problem #8.
>

You are right. But I just updated the page with maxima 5.28.0-2
and in all of them, the results were the same as Sage's, except for
#3. Sage did not evaluate it. Not sure why. Will be nice to know
which version of Maxima it is using. I use the notebook web
interface at http://www.sagenb.org

But Maxima on widnow, gave this result for #3

-----------------
integrate(asin(sqrt(x+1)-sqrt(x)),x);
(%pi*x)/2
-----------------

How did this come about? Did it just replace sqrt(x+1)-sqrt(x) with 1 ?

http://www.12000.org/my_notes/ten_hard_integrals/index.htm

--Nasser

[clicl...@freenet.de]

"Nasser M. Abbasi" schrieb:

>
> On 6/10/2013 5:23 AM, clicl...@freenet.de wrote:
>
> >
> > I strongly suspect that "Sage" always means "Maxima" (not e.g. "Sympy"):
> > see the Runtime Error Message for problem #8.
> >
>
> You are right. But I just updated the page with maxima 5.28.0-2
> and in all of them, the results were the same as Sage's, except for
> #3. Sage did not evaluate it. Not sure why. Will be nice to know
> which version of Maxima it is using. I use the notebook web
> interface at http://www.sagenb.org
>

It should be possible to switch the Sage integrator from "Maxima" to
"Sympy"; perhaps somebody could explain how? This would be preferable to
listing the Maxima results twice.

Martin.

[Nasser M. Abbasi]

On 6/13/2013 5:54 AM, clicl...@freenet.de wrote:

>
> It should be possible to switch the Sage integrator from "Maxima" to
> "Sympy";
> perhaps somebody could explain how? This would be preferable to
> listing the Maxima results twice.
>

Yes, ofcourse it would be. I'll try to find out how to do this
and if it is possible (I do not use Sage really at all, and
do not know much about how it works) and if I find out how to
make it use "Sympy" instead, will redo those intgrals using
that and update the listing.

thanks,
--Nasser

[Waldek Hebisch]

I tried sympy-0.7.2 on the ten. I got anwers for #2 and #10:

>>> integrate((x*asin(x))/sqrt(1 - x**2), x)
__________
/ 2
x - \/ - x + 1 *asin(x)
>>> integrate((x**3*exp(asin(x)))/sqrt(1 - x**2), x)
__________ __________
3 asin(x) 2 / 2 asin(x) asin(x) / 2 as
x *e 3*x *\/ - x + 1 *e 3*x*e 3*\/ - x + 1 *e
----------- - --------------------------- + ------------ - -------------------
10 10 10 10


in(x)

-----

#4 did not finish after 35 min. The other return unevaluated.
#9 took about 15 min to return unevaluated result, on most
other there was visible delay before answer.

Trying other examples which in principle sympy should do with
no trouble I had to wait few minutes in one case and in
another case sympy run out of memory after few hours.

--
Waldek Hebisch
heb...@math.uni.wroc.pl

[Nasser M. Abbasi]

On 6/13/2013 5:04 PM, Waldek Hebisch wrote:

>
> I tried sympy-0.7.2 on the ten. I got anwers for #2 and #10:

If any one interested, I have screen shots from sympy up now.

http://www.12000.org/my_notes/ten_hard_integrals/index.htm

I used whatever my Linux installed for me via package manager, which
it says it is sympy 0.7.1-rc1-3. It gave result for #10 for
this version.

>
> #4 did not finish after 35 min.

For me, it was #5 which never finished, waited 1 hr.

Also, sympy 0.7.1 did not know sec(). Must have been
added on version 0.7.2.

regards,
--Nasser

[Waldek Hebisch]

Nasser M. Abbasi <n...@12000.org> wrote:
> On 6/13/2013 5:04 PM, Waldek Hebisch wrote:
>
> >
> > I tried sympy-0.7.2 on the ten. I got anwers for #2 and #10:
>
> If any one interested, I have screen shots from sympy up now.
>
> http://www.12000.org/my_notes/ten_hard_integrals/index.htm
>
> I used whatever my Linux installed for me via package manager, which
> it says it is sympy 0.7.1-rc1-3. It gave result for #10 for
> this version.
>
> >
> > #4 did not finish after 35 min.
>
> For me, it was #5 which never finished, waited 1 hr.

I gave wrong number: it was #5. In fact it is still
runnig 2h 20min, 4.3GB RAM in use.

--
Waldek Hebisch
heb...@math.uni.wroc.pl

[clicl...@freenet.de]

Waldek Hebisch schrieb:

Better late than never :).

If my information is up-to-date, Sympy claims to field a complete
implementation the Risch procedures from Bronstein's volume 1, covering
everything elementary but non-algebraic, supplemented by an
implementation of the extended Risch-Norman heuristic (something like
Bronsteins "Poor Man's Integrator") for cases where these procedures do
not apply. Sympy also claims that it can automatically convert to and
determine the antiderivatives of Meijer G-functions.

So, what you are suffering here must be the "extended Risch-Norman
heuristic" mostly.

Thanks to you both for the Sympy demonstration!

Martin.

[Waldek Hebisch]

clicl...@freenet.de wrote:
>
> Waldek Hebisch schrieb:
> >
> > Nasser M. Abbasi <n...@12000.org> wrote:
> > > On 6/13/2013 5:04 PM, Waldek Hebisch wrote:
> > >
> > > >
> > > > I tried sympy-0.7.2 on the ten. I got anwers for #2 and #10:
> > >
> > > If any one interested, I have screen shots from sympy up now.
> > >
> > > http://www.12000.org/my_notes/ten_hard_integrals/index.htm
> > >
> > > I used whatever my Linux installed for me via package manager, which
> > > it says it is sympy 0.7.1-rc1-3. It gave result for #10 for
> > > this version.
> > >
> > > >
> > > > #4 did not finish after 35 min.
> > >
> > > For me, it was #5 which never finished, waited 1 hr.
> >
> > I gave wrong number: it was #5. In fact it is still
> > runnig 2h 20min, 4.3GB RAM in use.
> >
>
> Better late than never :).
>
> If my information is up-to-date, Sympy claims to field a complete
> implementation the Risch procedures from Bronstein's volume 1, covering
> everything elementary but non-algebraic,

I do not think so. IIUC they started working on such thing, but
it is far from finished.

> supplemented by an
> implementation of the extended Risch-Norman heuristic (something like
> Bronsteins "Poor Man's Integrator") for cases where these procedures do
> not apply.

AFAIK this is main integration procedure.

> Sympy also claims that it can automatically convert to and
> determine the antiderivatives of Meijer G-functions.

Yes, they have such a routine.

> So, what you are suffering here must be the "extended Risch-Norman
> heuristic" mostly.

Yes, that is the Risch-Norman part. I do not know why it is
so slow. Risch-Norman heuristic (with few improvements due
to Bronstein) implemented in FriCAS is much faster. The longest
running example is #9 needing 9s. FriCAS version can only do #2.
Risch-Norman heuristic can generate quite large systems of
linear equations, for example #9 generates system of 1814
equations in 794 unknowns. FriCAS version treats algebraics
like transcendentals, which means it is expected to fail
on integrals involving algebraics. Still, properly handling
algebraics should only moderately increase size of linear
systems (and sometimes may lead to smaller system).

--
Waldek Hebisch
heb...@math.uni.wroc.pl

[Richard Fateman]

Something like sympy can only be described at
best at a particular time/stage of development.
My guess is that the spectrum of what the integration
part does can be described as roughly this.

1. a stub of a program with name "integrate"
2. easy-to-do integrals for demo purposes mostly
3. more serious rational function methods added
4. partial implementation of Risch "algorithm"
5. retrenchment of easy-to-do integrals upon
realization that simplification, differential
algebra, and other algorithms matter
6. partial implementation of Meijer G-function
transformations.
7. Re-implementation of more of Risch "algorithm"
8. (for Mathematica) marketing claims of
doing everything in book X, even if false.
9. adding pattern matching with a vengeance ala Ruby.
10. Reorganization, go back to step (1).

Where in the spectrum the current program sits,
I don't know. It is of course possible that sympy
will have the best integration program there is.
But considering the amount of code it is alleged to
require in competing implementations, one wonders
how much effort will be needed to match it. Of course
if one of the Sage methods is "call Fricas and Maxima
and sympy in parallel and see what comes back" Sage
might be interesting, if kludgy.

[clicl...@freenet.de]

Waldek Hebisch schrieb:

>
> clicl...@freenet.de wrote:
> >
> > If my information is up-to-date, Sympy claims to field a complete
> > implementation the Risch procedures from Bronstein's volume 1,
> > covering everything elementary but non-algebraic,
>
> I do not think so. IIUC they started working on such thing, but
> it is far from finished.
>

You are right. On googling a bit I found this description by Aaron
Meurer in a March 2013 post:

<http://www.mail-archive.com/sy...@googlegroups.com/msg17613.html>

"... I can tell you from Bronstein's book:
- everything before chapter 5 is preliminaries, and was basically
implemented before I even started.
- chapter 5 is the base of the algorithm, and has all been implemented,
except for 5.12.
- most of chapter 6 is implemented. The exceptions are parts that
require certain subalgorithms that haven't been implemented yet, namely
the stuff from 5.12.
- only a small part of chapter 7 is implemented.
- none of chapter 8 is implemented.
- the parts of chapter 9 relating to exponentials and logarithms have
been implemented."

Martin.

[clicl...@freenet.de]

clicl...@freenet.de schrieb:
>
> here are my antiderivatives for problems #1 to #10, as promised. I
> think they are as compact, continuous, real, and elementary as one
> could wish them to be:
>
> [...]
>
> INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> = x/3 + 1/3*ATAN(SIN(x)*COS(x)*(COS(x)^2 + 1)
> /(COS(x)^2*SQRT(COS(x)^4*COS(x)^2 + 1) + 1))
>

Oops, this should have read:

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

Another simple elementarization of the elliptic problem #49 from
Charlwood's appendix is:

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

Martin.

[Nasser M. Abbasi]

On 6/16/2013 11:30 AM, clicl...@freenet.de wrote:
>

>
> Oops, this should have read:
>
> INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> = x/3 + 1/3*ATAN(SIN(x)*COS(x)*(COS(x)^2 + 1)
> /(COS(x)^2*SQRT(COS(x)^4 + COS(x)^2 + 1) + 1))
>

fyi, for the above, which #5, the LeafCount is 45

---------------------------------------

f1= x/3+1/3*ArcTan[Sin[x]*Cos[x]*(Cos[x]^2+1)/(Cos[x]^2*Sqrt[Cos[x]^4+Cos[x]^2+1]+1)];

LeafCount[f1]
-----------------------------------------
45

For the one listed in the table, it is 37:

------------------------------------------

f0 =(-ArcSin[Cos[x]^3]*Sqrt[1-Cos[x]^6]*Csc[x])/(3*Sqrt[1+Cos[x]^2+Cos[x]^4]);

LeafCount[f0]

Out[13]= 37
---------------------------------------

I was wondering: assuming both antiderivates contain
just elemetary functions, can one then use the leaf count as a
measure of which is most optimal answer? Or will there
be other considerations one should look at or better
way to measure which one is more "optimal" than the
other?

reference:
http://reference.wolfram.com/mathematica/ref/LeafCount.html

thanks,
--Nasser

[clicl...@freenet.de]

"Nasser M. Abbasi" schrieb:

Apart from the compactness of antiderivatives, as measured by leaf
counting, continuity on the real axis and absence of complex
intermediate results when evaluated on the real axis (which implies
absence of the imaginary unit) are important in my view, and usually
take precedence over compactness.

Thus, my 45-leafed result is fully continuous along the real axis,
whereas the shorter ATAN alternative:

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

= - 1/3*ATAN(COT(x)*COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1))

as well as Albert's 37-leafed ASIN version:

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

= - ASIN(COS(x)^3)*SQRT(1 - COS(x)^6)*CSC(x)
/(3*SQRT(1 + COS(x)^2 + COS(x)^4))

jump at x = -pi, 0, pi, etc. This constitutes an unnecessary obstacle in
definite integration - imagine some quantity integrated along the path
of an orbiting spacecraft.

I usually accept logarithmic evaluations like INT(1/x, x) = LN(x), which
can be complex where the integrand is real (here for x < 0). I think
that users (e.g. calculus students) who need this integral from x = -2
to x = -1, say, should be able to accept that constants involving some
formal quantity #i appear which drop out of the final result.

Martin.

[Nasser M. Abbasi]

On 6/17/2013 11:11 AM, clicl...@freenet.de wrote:
>
> Apart from the compactness of antiderivatives, as measured by leaf
> counting, continuity on the real axis and absence of complex
> intermediate results when evaluated on the real axis (which implies
> absence of the imaginary unit) are important in my view, and usually
> take precedence over compactness.
>
> Thus, my 45-leafed result is fully continuous along the real axis,
> whereas the shorter ATAN alternative:
>
> INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> = - 1/3*ATAN(COT(x)*COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1))
>
> as well as Albert's 37-leafed ASIN version:
>
> INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> = - ASIN(COS(x)^3)*SQRT(1 - COS(x)^6)*CSC(x)
> /(3*SQRT(1 + COS(x)^2 + COS(x)^4))
>
> jump at x = -pi, 0, pi, etc. This constitutes an unnecessary obstacle in
> definite integration - imagine some quantity integrated along the path
> of an orbiting spacecraft.
>

I noticed that last night when I made a plot of them to compare. Here is
the plot

http://12000.org/tmp/061713/no_5.png

I might add a link then next to each given optimal entry in
the table showing a plot of the antiderivate, will be easy to add.

> I usually accept logarithmic evaluations like INT(1/x, x) = LN(x), which
> can be complex where the integrand is real (here for x < 0). I think
> that users (e.g. calculus students) who need this integral from x = -2
> to x = -1, say, should be able to accept that constants involving some
> formal quantity #i appear which drop out of the final result.
>
> Martin.
>

Thanks for the information, this helped.

On a related point, would you please help me understand how
free version of reduce transformed

arcsin(x)*log(x)

to

arcsin( sin(g0) ) * cos(g0) * log( sin(g0) )

by replacing x with sin(g0).

i.e Where does cos(g0) term come from in the above transformation?

Here is a link to the reduce trace for integral #1, which it
could not do btw. And the above was the first step in the process.

http://12000.org/my_notes/ten_hard_integrals/reduce_logs/1/HTML/trace_1.html

thanks,
--Nasser

[clicl...@freenet.de]

"Nasser M. Abbasi" schrieb:

>
> On a related point, would you please help me understand how
> free version of reduce transformed
>
> arcsin(x)*log(x)
>
> to
>
> arcsin( sin(g0) ) * cos(g0) * log( sin(g0) )
>
> by replacing x with sin(g0).
>
> i.e Where does cos(g0) term come from in the above transformation?
>
> Here is a link to the reduce trace for integral #1, which it
> could not do btw. And the above was the first step in the process.
>
> http://12000.org/my_notes/ten_hard_integrals/reduce_logs/1/HTML/trace_1.html
>

In the original INT(ASIN(x)*LN(x), x), Reduce makes the variable
substitution x = SIN(y), dx = COS(y)*dy (I've written y for g0). So the
factor COS(y) just represents the derivative dx/dy that must be included
in the transformed integrand.

It looks like the integration process is simply restarted after the
variable substitution. While the first step is easy to understand, I
have no clear idea what Reduce is trying to do later - and without
success. It may be the Risch-Norman heuristic again.

By the way, Derive 6.10 attacks the original integral using integration
by parts. It can also evaluate INT(x*COS(x)*LN(SIN(x)), x) which is
obtained when ASIN(SIN(x)) in the variable-substituted integral is
replaced by x (this replacement is not valid for all complex x, nor for
all real x, however).

Martin.

[Nasser M. Abbasi]

On 6/17/2013 5:09 PM, clicl...@freenet.de wrote:
>
> "Nasser M. Abbasi" schrieb:
>>
>> On a related point, would you please help me understand how
>> free version of reduce transformed
>>
>> arcsin(x)*log(x)
>>
>> to
>>
>> arcsin( sin(g0) ) * cos(g0) * log( sin(g0) )
>>
>> by replacing x with sin(g0).
>>
>> i.e Where does cos(g0) term come from in the above transformation?
>>
>> Here is a link to the reduce trace for integral #1, which it
>> could not do btw. And the above was the first step in the process.
>>
>> http://12000.org/my_notes/ten_hard_integrals/reduce_logs/1/HTML/trace_1.html
>>


>
> In the original INT(ASIN(x)*LN(x), x), Reduce makes the variable
> substitution x = SIN(y), dx = COS(y)*dy (I've written y for g0). So the
> factor COS(y) just represents the derivative dx/dy that must be included
> in the transformed integrand.
>

Ah! Yes, ofcourse, but the trace was confusing. If you see, it says
it was transforming arcsin(x)*log(x) and NOT arcsin(x)*log(x)*dx

If dx is included, then it works out. Ok, here it is step
by step again.

given:
I = arcsin(x) log(x) dx

let x=sin(y), then dx/dy=cos(y), or dx=cos(y)*dy
hence I becomes

arcsin(sin(y)) log(sin(y) cos(y) dy

So it was the change from dx to dy which introduced the cos(y)
term, that is what I overlooked since trace did not show dx in
what it started with.

thanks
--Nasser

[Waldek Hebisch]

I am not sure why Reduce fares so poorly on this problem. It
can do:

3: int(x*cos(x)*log(sin(x)), x);

x
2*tan(---)
2 x 2
cos(x)*log(---------------) - 2*cos(x) - log(tan(---) + 1)
x 2 2
tan(---) + 1
2

x x
2*tan(---) 2*tan(---)
2 2
+ log(---------------)*sin(x)*x - log(---------------) - sin(x)*x + 2
x 2 x 2
tan(---) + 1 tan(---) + 1
2 2

so maybe it fails at backsubstitution x -> asin(x). Or maybe
it got doubts like you against replacing asin(sin(x)) by x.
But such replacement is fine:

(21) -> integrate(x*cos(x)*log(sin(x)), x)

(21)
cos(x) + 1 - cos(x) + 1
(2x sin(x) + 2cos(x))log(sin(x)) + log(----------) - log(------------)
2 2
+
- 2x sin(x) - 4cos(x)
/
2
Type: Union(Expression(Integer),...)
(22) -> eval(%, x = asin(x))

(22)
+--------+ +--------+
| 2 | 2 +--------+
\|- x + 1 + 1 - \|- x + 1 + 1 | 2
log(---------------) - log(-----------------) + (2log(x) - 4)\|- x + 1
2 2
+
2x asin(x)log(x) - 2x asin(x)
/
2
Type: Expression(Integer)
(23) -> ii := D(%, x)

(23) asin(x)log(x)
Type: Expression(Integer)

In general, since original integrand is analytic except for {-1, 0, 1},
it is enough for substitution to be valid in neighbourhood of a
single point. You just need to be careful to choose desired
branch after backsubstition.

--
Waldek Hebisch
heb...@math.uni.wroc.pl

[Albert Rich]

On Monday, June 17, 2013 6:11:36 AM UTC-10, clicl...@freenet.de wrote:

> > On 6/16/2013 11:30 AM, clicl...@freenet.de wrote:

> > > Oops, [problem #5] should have read:

>
> > > INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> > > = x/3 + 1/3*ATAN(SIN(x)*COS(x)*(COS(x)^2 + 1)
> > > /(COS(x)^2*SQRT(COS(x)^4 + COS(x)^2 + 1) + 1))
>

> Apart from the compactness of antiderivatives, as measured by leaf
> counting, continuity on the real axis and absence of complex
> intermediate results when evaluated on the real axis (which implies
> absence of the imaginary unit) are important in my view, and usually
> take precedence over compactness.
>
> Thus, my 45-leafed result is fully continuous along the real axis,
> whereas the shorter ATAN alternative:
>
> INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> = - 1/3*ATAN(COT(x)*COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1))
>
> as well as Albert's 37-leafed ASIN version:
>
> INT(COS(x)^2/SQRT(COS(x)^4 + COS(x)^2 + 1), x) =
> = - ASIN(COS(x)^3)*SQRT(1 - COS(x)^6)*CSC(x)
> /(3*SQRT(1 + COS(x)^2 + COS(x)^4))
>
> jump at x = -pi, 0, pi, etc. This constitutes an unnecessary obstacle in
> definite integration - imagine some quantity integrated along the path
> of an orbiting spacecraft.

I agree continuity of antiderivatives trumps compactness. Your impressive, continuous result for problem #5 is in the revised Charlwood Fifty pdf file at

http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf

Please let me know if other problems can be continuitized, to coin a phrase.

Albert

[clicl...@freenet.de]

A minor observation: The first term of the solution to problem #35 from
Charlwood's appendix should be simplified (that is, the square root in
the denominator be removed).

<http://www.apmaths.uwo.ca/~arich/CharlwoodIntegrationProblems.pdf>

Martin.

[Albert Rich]

A revised pdf file has been posted. Thanks.

Albert