
Re: The Charlwood Fifty
Posted:
May 23, 2013 12:17 PM


Albert Rich schrieb: > > 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 testsuite, 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 testsuite, 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
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) pseudoelliptic:
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 pseudoelliptic 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 à Condorcet of May 1778, reprinted in the Bulletin des Sciences Mathématiques et Astronomiques (2e série), tome 3, no 1 (1879), p. 225227:
<http://www.numdam.org/item?id=BSMA_1879_2_3_1_225_0>
Martin.

