Albert Rich schrieb:   
   >   
   > On Monday, January 22, 2018 at 3:29:26 AM UTC-10, clicl...@freenet.de wrote:   
   > >   
   > > I don't think the example on which your rule 482 is based was provided   
   > > by me (no such example was in my files until very recently), but perhaps   
   > > you transformed an example containing (1 - x^2)^(1/3) by making this   
   > > radical the new integration variable. Anyway, the following result   
   > > should be much simpler than the elementary antiderivative returned by   
   > > FriCAS:   
   > >   
   > > INT(18*x/((x^3 + 8)*SQRT(x^3 - 1)), x)   
   > >  = SQRT(3)*ATANH(SQRT(3)*(x - 1)/SQRT(x^3 - 1))   
   > >  - ATAN(3/SQRT(x^3 - 1))   
   > >  + ATAN((x - 1)^2/(3*SQRT(x^3 - 1)))   
   > >   
   > > where (if you prefer):   
   > >   
   > > - ATAN((x - 1)^2/(3*SQRT(x^3 - 1)))   
   > >  = ATAN((x - 1)/SQRT(x^3 - 1)) + ATAN((2*x + 1)/SQRT(x^3 - 1))   
   > >   
   >   
   > Thanks! I generalized the above result to integrate expressions of the   
   > form x/((a+b*x^3)*sqrt(c+d*x^3)) when 8*b*c+a*d=0 and c is positive   
   > and when c is negative.  The forthcoming Rubi 4.14.5 will incorporate   
   > these two new rules, as well as two corresponding rules for   
   > expressions of the form 1/((a+b*x^2)^(1/3)*(c+d*x^2)) when   
   > b*c-9*a*d=0.   
      
   However, the integrand 1/((3 + x^2)*(1 + 3*x^2)^(1/3)) has appeared on   
   sci.math.symbolic, e.g. on 25th August 2017 in the thread "multiple   
   non-linear factors & trace zero". Euler's paper is entitled "Integratio   
   Succincta Formulae Integralis Maxime Memorabilis ..." and appeared in   
   the Nova Acta Acad. Imp. Scient. Tom. X (1792) pp. 20-26.   
      
   >   
   > However, for your first example integrand x/((4-x^3)*sqrt(1-x^3)), I   
   > have tried in vain to discover a simple antiderivative involving just   
   > 3 or 4 arctan(h) and/or log terms. If you or someone derives an   
   > antiderivative simpler than the one Rubi currently returns, please let   
   > me know. Rubi will appreciate it.   
   >   
      
   In my files I have:   
      
   INT(x/((4 - x^3)*SQRT(1 - x^3)), x)   
    = 2^(1/3)/18*(ATANH(SQRT(1 - x^3))   
    - 3*ATANH((1 + 2^(1/3)*x)/SQRT(1 - x^3))   
    - SQRT(3)*ATAN((2^(1/3) - 2^(2/3)*x - x^2)   
                   /(SQRT(3)*2^(1/3)*SQRT(1 - x^3))))   
      
   where   
      
   ATAN((2^(1/3) - 2^(2/3)*x - x^2)/(SQRT(3)*2^(1/3)*SQRT(1 - x^3)))   
    = ATAN(1/SQRT(3) - 2^(2/3)*(1 - SQRT(1 - x^3))/(SQRT(3)*x))   
    - ATAN(1/SQRT(3) - 2^(2/3)*(1 + SQRT(1 - x^3))/(SQRT(3)*x))   
    = ATAN(SQRT(3)/SQRT(1 - x^3))   
    + ATAN(SQRT(3)*(1 - 2^(1/3)*x)/SQRT(1 - x^3))   
      
   Unfortunately, the ATANH combination is discontinuous at x = 0; I could   
   not improve this defect so far.   
      
   Martin.   
      
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