From: nma%12000.org@gtempaccount.com   
      
   On Monday, November 13, 2023 at 5:49:22 PM UTC-6, Sam Blake wrote:   
   > Hi All,    
   >    
   > Is there any way to a priori estimate of the time it will take the   
   Risch-Trager-Bronstein algorithm to compute the integral of    
   >    
   > (1 - x^4)/((x^4 + x^2 + 1)*(x^5 - x^3)^(1/4))?    
   >    
   > I left it for 30 minutes and it was still crunching away...    
      
   On what system do you mean? in V 13.3 Mathematica solves it in 2.3 seconds   
      
   In[1]:= Integrate[(1-x^4)/((x^4+x^2+1)*(x^5-x^3)^(1/4)),x]//Timing   
   Out[1]= {2.375,(1/(2^(3/4) (51+36 Sqrt[2])^(1/8) (x^3 (-1+x^2))^(1/4)))x^(3/4)   
   (-1+x^2)^(1/4) (ArcTan[(2^(1/4) 3^(7/8) x^(1/4) (-1+x^2)^(1/4))/((17+12   
   Sqrt[2])^(1/8) (-3 Sqrt[x]+3^(3/4) Sqrt[-1+x^2]))]+(17+12 Sqrt[2])^(1/4)   
   ArcTan[(2^(1/4) 3^(7/8) (17+12    
   Sqrt[2])^(1/8) x^(1/4) (-1+x^2)^(1/4))/(-3 Sqrt[x]+3^(3/4) Sqrt[   
   1+x^2])]+ArcTanh[(2^(1/4) 3^(7/8) x^(1/4) (-1+x^2)^(1/4))/((17+12   
   Sqrt[2])^(1/8) (3 Sqrt[x]+3^(3/4) Sqrt[-1+x^2]))]+(17+12 Sqrt[2])^(1/4)   
   ArcTanh[(2^(1/4) 3^(7/8) (17+12 Sqrt[2])^(1/8) x^(1/   
   4) (-1+x^2)^(1/4))/(3 Sqrt[x]+3^(3/4) Sqrt[-1+x^2])])}   
      
   Rubi 4.17.3 solves it in 0.35 seconds, but in terms of special functions   
      
   In[3]:= Int[(1-x^4)/((x^4+x^2+1)*(x^5-x^3)^(1/4)),x]//Timing   
   Out[3]= {0.359375,(4 (3 I+Sqrt[3]) x (1-x^2)^(1/4) AppellF1[1/8,   
   (3/4),1,9/8,x^2,-((2 x^2)/(1-I Sqrt[3]))])/(3 (I+Sqrt[3]) (-x^3+x^5)^(1/4))+(4   
   (3 I-Sqrt[3]) x (1-x^2)^(1/4) AppellF1[1/8,-(3/4),1,9/8,x^2,-((2 x^2)/(1+I   
   Sqrt[3]))])/(3 (I-Sqrt[3]) (-x^3+x^   
   5)^(1/4))}   
      
   >    
   > It takes my package 24 seconds on my old laptop    
   >    
   > In[22299]:= IntegrateAlgebraic[(1 - x^4)/((x^4 + x^2 + 1) (x^5 -   
   x^3)^(1/4)), x] // Timing    
   >    
   > Out[22299]= {23.9914, -(((1/3 (17 + 12 Sqrt[2]))^(1/8)    
   > ArcTan[(    
   > 3 Sqrt[1 + Sqrt[2]] x)/(-3 Sqrt[1 + Sqrt[2]] x +    
   > 3 Sqrt[2 + 2 Sqrt[2]] x - 2^(3/4) 3^(7/8) (-x^3 + x^5)^(1/4))])/    
   > 2^(3/4)) + ((1/3 (17 + 12 Sqrt[2]))^(1/8)    
   > ArcTan[(    
   > 3 Sqrt[1 + Sqrt[2]] x)/(-3 Sqrt[1 + Sqrt[2]] x +    
   > 3 Sqrt[2 + 2 Sqrt[2]] x + 2^(3/4) 3^(7/8) (-x^3 + x^5)^(1/4))])/    
   > 2^(3/4) -    
   > ArcTan[(-2^(1/4) (3/(17 + 12 Sqrt[2]))^(1/8) x^2 + (-3 x^2 +    
   > 3^(3/4) (17 + 12 Sqrt[2])^(1/4) Sqrt[-x^3 + x^5])/(    
   > 2^(1/4) 3^(7/8) (17 + 12 Sqrt[2])^(1/8)))/(    
   > x (-x^3 + x^5)^(1/4))]/(2^(3/4) (3 (17 + 12 Sqrt[2]))^(1/8)) +    
   > ArcTanh[(    
   > 3 (17/8748 + Sqrt[2]/729)^(1/8) x^2 +    
   > 3^(3/4) (17/8748 + Sqrt[2]/729)^(1/8) Sqrt[-x^3 + x^5])/(    
   > x (-x^3 + x^5)^(1/4))]/(    
   > 2^(3/4) (3 (17 + 12 Sqrt[2]))^(1/8)) + ((1/3 (17 + 12 Sqrt[2]))^(    
   > 1/8) ArcTanh[(((3/(17 + 12 Sqrt[2]))^(1/8) x^2)/2^(1/4) +    
   > Sqrt[-x^3 + x^5]/(2^(1/4) (3 (17 + 12 Sqrt[2]))^(1/8)))/(    
   > x (-x^3 + x^5)^(1/4))])/2^(3/4)}    
   >    
   > Cheers,    
   >    
   > Sam   
      
   Fricas 1.3.9 does it in less than 2 minutes and gives this   
      
   (3) -> ii:=integrate((1 - x^4)/((x^4 + x^2 + 1)*(x^5 - x^3)^(1/4)),x);   
   (4) -> unparse(ii::InputForm)   
      
     "((-2)*(-1)^(1/2)*(-1)^(1/8)*(3^(1/8))^7*log((((264*x^2+654*x+(-264))*((-1)^(   
     1/8))^4*(3^(1/8))^4+((-654)*x^2+792*x+654))*((x^5+(-1)*x^3)^(1/4))^3+((264*x^   
     3+654*x^2+(-264)*x)*(-1)^(1/2)*((-1)^(1/8))^3*(3^(1/8))^5+(654*x^3+(-792)*x^2   
     +(-654)*x)*(-1)^(1/2)*((-1)^(1/8))^7*3^(1/8))*((x^5+(-1)*x^3)^(1/4))^2+(((-26   
     4)*x^4+(-654)*x^3+264*x^2)*((-1)^(1/8))^2*(3^(1/8))^6+((-654)*x^4+792*x^3+654   
     *x^2)*((-1)^(1/8))^6*(3^(1/8))^2)*(x^5+(-1)*x^3)^(1/4)+((109*x^6+(-264)*x^5+(   
     -545)*x^4+264*x^3+109*x^2)*(-1)^(1/2)*(-1)^(1/8)*(3^(1/8))^7+((-132)*x^6+(-65   
     4)*x^5+660*x^4+654*x^3+(-132)*x^2)*(-1)^(1/2)*((-1)^(1/8))^5*(3^(1/8))^3))/(x   
     ^6+x^4+x^2))+(2*(-1)^(1/8)*(3^(1/8))^7*log((((264*x^2+654*x+(-264))*((-1)^(1/   
     8))^4*(3^(1/8))^4+((-654)*x^2+792*x+654))*((x^5+(-1)*x^3)^(1/4))^3+((264*x^3+   
     654*x^2+(-264)*x)*((-1)^(1/8))^3*(3^(1/8))^5+(654*x^3+(-792)*x^2+(-654)*x)*((   
     -1)^(1/8))^7*3^(1/8))*((x^5+(-1)*x^3)^(1/4))^2+((264*x^4+654*x^3+(-264)*x^2)*   
     ((-1)^(1/8))^2*(3^(1/8))^6+(654*x^4+(-792)*x^3+(-654)*x^2)*((-1)^(1/8))^6*(3^   
     (1/8))^2)*(x^5+(-1)*x^3)^(1/4)+(((-109)*x^6+264*x^5+545*x^4+(-264)*x^3+(-109)   
     *x^2)*(-1)^(1/8)*(3^(1/8))^7+(132*x^6+654*x^5+(-660)*x^4+(-654)*x^3+132*x^2)*   
     ((-1)^(1/8))^5*(3^(1/8))^3))/(x^6+x^4+x^2))+((-2)*(-1)^(1/8)*(3^(1/8))^7*log(   
     (((264*x^2+654*x+(-264))*((-1)^(1/8))^4*(3^(1/8))^4+((-654)*x^2+792*x+654))*(   
     (x^5+(-1)*x^3)^(1/4))^3+(((-264)*x^3+(-654)*x^2+264*x)*((-1)^(1/8))^3*(3^(1/8   
     ))^5+((-654)*x^3+792*x^2+654*x)*((-1)^(1/8))^7*3^(1/8))*((x^5+(-1)*x^3)^(1/4)   
     )^2+((264*x^4+654*x^3+(-264)*x^2)*((-1)^(1/8))^2*(3^(1/8))^6+(654*x^4+(-792)*   
     x^3+(-654)*x^2)*((-1)^(1/8))^6*(3^(1/8))^2)*(x^5+(-1)*x^3)^(1/4)+((109*x^6+(-   
     264)*x^5+(-545)*x^4+264*x^3+109*x^2)*(-1)^(1/8)*(3^(1/8))^7+((-132)*x^6+(-654   
     )*x^5+660*x^4+654*x^3+(-132)*x^2)*((-1)^(1/8))^5*(3^(1/8))^3))/(x^6+x^4+x^2))   
     +(2*(-1)^(1/2)*(-1)^(1/8)*(3^(1/8))^7*log((((264*x^2+654*x+(-264))*((-1)^(1/8   
     ))^4*(3^(1/8))^4+((-654)*x^2+792*x+654))*((x^5+(-1)*x^3)^(1/4))^3+(((-264)*x^   
     3+(-654)*x^2+264*x)*(-1)^(1/2)*((-1)^(1/8))^3*(3^(1/8))^5+((-654)*x^3+792*x^2   
     +654*x)*(-1)^(1/2)*((-1)^(1/8))^7*3^(1/8))*((x^5+(-1)*x^3)^(1/4))^2+(((-264)*   
     x^4+(-654)*x^3+264*x^2)*((-1)^(1/8))^2*(3^(1/8))^6+((-654)*x^4+792*x^3+654*x^   
     2)*((-1)^(1/8))^6*(3^(1/8))^2)*(x^5+(-1)*x^3)^(1/4)+(((-109)*x^6+264*x^5+545*   
     x^4+(-264)*x^3+(-109)*x^2)*(-1)^(1/2)*(-1)^(1/8)*(3^(1/8))^7+(132*x^6+654*x^5   
     +(-660)*x^4+(-654)*x^3+132*x^2)*(-1)^(1/2)*((-1)^(1/8))^5*(3^(1/8))^3))/(x^6+   
     x^4+x^2))+(((-1)^(1/2)+(-1))*(-1)^(1/8)*2^(1/2)*(3^(1/8))^7*log(((((-528)*x^2   
     +(-1308)*x+528)*((-1)^(1/8))^4*(3^(1/8))^4+((-1308)*x^2+1584*x+1308))*((x^5+(   
     -1)*x^3)^(1/4))^3+(((264*x^3+654*x^2+(-264)*x)*(-1)^(1/2)+(264*x^3+654*x^2+(-   
     264)*x))*((-1)^(1/8))^3*2^(1/2)*(3^(1/8))^5+(((-654)*x^3+792*x^2+654*x)*(-1)^   
     (1/2)+((-654)*x^3+792*x^2+654*x))*((-1)^(1/8))^7*2^(1/2)*3^(1/8))*((x^5+(-1)*   
     x^3)^(1/4))^2+(((-528)*x^4+(-1308)*x^3+528*x^2)*(-1)^(1/2)*((-1)^(1/8))^2*(3^   
     (1/8))^6+(1308*x^4+(-1584)*x^3+(-1308)*x^2)*(-1)^(1/2)*((-1)^(1/8))^6*(3^(1/8   
     ))^2)*(x^5+(-1)*x^3)^(1/4)+((((-109)*x^6+264*x^5+545*x^4+(-264)*x^3+(-109)*x^   
     2)*(-1)^(1/2)+(109*x^6+(-264)*x^5+(-545)*x^4+264*x^3+109*x^2))*(-1)^(1/8)*2^(   
     1/2)*(3^(1/8))^7+(((-132)*x^6+(-654)*x^5+660*x^4+654*x^3+(-132)*x^2)*(-1)^(1/   
     2)+(132*x^6+654*x^5+(-660)*x^4+(-654)*x^3+132*x^2))*((-1)^(1/8))^5*2^(1/2)*(3   
     ^(1/8))^3))/(x^6+x^4+x^2))+(((-1)^(1/2)+1)*(-1)^(1/8)*2^(1/2)*(3^(1/8))^7*log   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)