clicliclic@freenet.de  wrote:   
   >   
   > To solve a sweet little algebraic integral:   
   >   
   > integrate(1/((x + 1)*(x^3 + 2)^(1/3)), x)   
   >   
   > FriCAS 1.3.5 invokes monstrous coefficients:   
      
   > A compact version of the antiderivative reads:   
   >   
   > INT(1/((x + 1)*(x^3 + 2)^(1/3)), x)   
   >  = 1/12*(- 3*LN((x^3 + 2)^(1/3) - x)   
   >  + 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*x/(x^3 + 2)^(1/3))))   
   >  - 1/4*(LN((x + 2)^3 - (x^3 + 2))   
   >  - 3*LN((x + 2) - (x^3 + 2)^(1/3))   
   >  + 2*SQRT(3)*ATAN(1/SQRT(3)*(1 + 2*((x + 2)/(x^3 + 2)^(1/3)))))   
   >   
   > The FriCAS antiderivative is correct, but why tripling the second ATAN   
   > and adding it to the first and then rationalizing the denominator   
   > should require 30-digit integers remains a mystery. Merging the above   
   > ATANs just gives:   
   >   
   > SQRT(3)/6*ATAN(SQRT(3)   
   > *(x^3 + 2)^(1/3)*((x^3 + 2)^(1/3)*(x*(x + 2)*(x^3 + 2)^(1/3)   
   >  + x^3 + 2*x^2 - 2) + x^4 + 2*x^3 + 2*x + 4)   
   > /((x^3 + 2)^(1/3)*((x^3 + 2)^(1/3)*(x^3 + 6*x^2 + 8*x + 2)   
   >  + x^4 + 6*x^3 + 8*x^2 + 6*x + 4)   
   >  + x^5 + 6*x^4 + 8*x^3 + 2*x^2 + 12*x + 16))   
   >   
   > FriCAS should probably try to simplify its ATANs by adding a constant   
   > ATAN that involves the same algebraic constants. Is this an ill-posed   
   > problem?   
      
   Internally FriCAS works with logarithms, so it gets   
      
   (-1/6)*(c1*log(f1) + c2*log(f2))   
      
   where c1 and c2 are two roots of c^2 + c + 1 = 0.  f1 and f2   
   are rather complicated, but are essentially unique for given   
   ci-s.  Quite possible that different choice of ci-s could   
   get simpler expression, but IFAIK there are no methods to   
   chose good ci-s except for trying several possibilities.   
   Each trial leads to rather expensive computation, so   
   FriCAS just tries once.  Note: if that trial fails, then   
   there is no elementary integral.  Other trials would be   
   just to see if we can get simpler expression.  Heuristically   
   we can expect that most alternative trials would produce   
   more complex expressions, increasing runtime.   
      
   Anyway, one factor leading to large result is suboptimal   
   choice of constants.  Another factor is transformation to   
   ATAN-s.  Namely, given sqrt(-1), ATAN can be transformed   
   to logarithm in rational way.  But going from logarithms   
   to ATAN-s may lead to doubling.  ATM I did not work out   
   when doubling is avoidable, but your examples shows that   
   there are many cases of avoidable doubling.   
      
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