From: samuel.thomas.blake@gmail.com   
      
   On Saturday, January 13, 2024 at 8:29:08 PM UTC+11, nob...@nowhere.invalid   
   wrote:   
   > Sam Blake schrieb:   
   > >    
   > > On Saturday, January 13, 2024 at 1:56:38 PM UTC+11, Sam Blake wrote:    
   > > >    
   > > > ... and here's an example where Rubi outperforms both Mathematica    
   > > > and my implementation of Mack's linear Hermite reduction (as given    
   > > > in Symbolic Integration I by Manuel Bronstein)    
   > > >    
   > > > In[1151]:= (2 - 9 x^4 + 2 Sqrt[2] x^4 - 12 x^6 - 3 x^8)/(Sqrt[2] - 3 x^2   
   - x^4)^3;    
   > > > HermiteReduce[%, x]    
   > > > Integrate[%%, x]    
   > > > Int[%%%, x]    
   > > >    
   > > > Out[1152]= {-((37549921350591017222596 (102743992 x +    
   > > > 72655641 Sqrt[2] x))/((-8 + 9 Sqrt[2]) (-113 +    
   > > > 72 Sqrt[2])^4 (2200 + 1593 Sqrt[2])^2 (-32544 +    
   > > > 23137 Sqrt[2]) (3352883803641 +    
   > > > 2370846461804 Sqrt[2]) (-3186 - 2200 Sqrt[2] + 6600 x^2 +    
   > > > 4779 Sqrt[2] x^2 + 2200 x^4 + 1593 Sqrt[2] x^4))), 0}    
   > > >    
   > > > Out[1153]= -(1/(    
   > > > 105 (-2 + 3 Sqrt[2] x^2 + Sqrt[2] x^4)^3))(Sqrt[2] - 3 x^2 -    
   > > > x^4)^3 (210 x AppellF1[1/2, 3, 3, 3/2, (    
   > > > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(    
   > > > 3 + Sqrt[9 + 4 Sqrt[2]]))] +    
   > > > 21 (-9 + 2 Sqrt[2]) x^5 AppellF1[5/2, 3, 3, 7/2, (    
   > > > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(    
   > > > 3 + Sqrt[9 + 4 Sqrt[2]]))] -    
   > > > 5 (36 x^7 AppellF1[7/2, 3, 3, 9/2, (    
   > > > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(    
   > > > 3 + Sqrt[9 + 4 Sqrt[2]]))] +    
   > > > 7 x^9 AppellF1[9/2, 3, 3, 11/2, (    
   > > > 2 x^2)/(-3 + Sqrt[9 + 4 Sqrt[2]]), -((2 x^2)/(    
   > > > 3 + Sqrt[9 + 4 Sqrt[2]]))]))    
   > > >    
   > > > Out[1154]= x/(Sqrt[2] - 3 x^2 - x^4)    
   > >    
   > > For fairness, Rubi makes a mess of this one    
   > >    
   > > In[1578]:= Int[(-2 - 2*Sqrt[3]*x - 3*x^2 + 4*Sqrt[3]*x^3 + 7*x^4 +   
   6*Sqrt[3]*x^5 + 10*x^6 + 2*Sqrt[3]*x^7 + 6*x^8 + x^10)/(1 + 3*x^2 +   
   Sqrt[3]*x^3 + 3*x^4 + x^6)^2, x]    
   > >    
   > > Out[1578]= -((3*3^(1/6) + 4*Sqrt[3] -    
   > > 5*3^(5/6) + (3 - 4*3^(2/3))*    
   > > x)/(3*(3 - 4*3^(2/3))*(1 + 3^(1/6)*x + x^2))) -    
   > > (3^(1/3)*(6*I - 3*(-1)^(2/3)*3^(1/6) + 2*Sqrt[3] +    
   > > 5*3^(5/6) + (-1)^(1/3)*(3 + 3^(2/3)*(2 + 2*I*Sqrt[3]))*x))/    
   > > ((1 + (-1)^(1/3))^4*(4 - (-1)^(2/3)*3^(1/3))*(1 - (-1)^(1/    
   > > 3)*3^(1/6)*x + x^2)) -    
   > > (2*(-1)^(1/3)*(3*(-1)^(2/3)*3^(1/6) + 4*Sqrt[3] +    
   > > 5*(-1)^(1/3)*3^(5/6) + (3 - 4*(-3)^(2/3))*x))/    
   > > (3*3^(2/3)*(8 + 3^(1/3) + I*3^(5/6))*(1 + (-1)^(2/3)*3^(1/6)*x +    
   > > x^2)) +    
   > > (2*ArcTan[((-1)^(1/3)*3^(1/6) - 2*x)/    
   > > Sqrt[4 - (-1)^(2/3)*3^(1/3)]])/(3*    
   > > Sqrt[(1/2)*(8 + 3^(1/3) - I*3^(5/6))]) -    
   > > (2*ArcTan[((-1)^(2/3)*3^(1/6) + 2*x)/Sqrt[4 + (-3)^(1/3)]])/(3*    
   > > Sqrt[(1/2)*(8 + 3^(1/3) + I*3^(5/6))]) +    
   > > (2*I*ArcTanh[(3^(1/6)*(I - Sqrt[3]) - 4*I*x)/    
   > > Sqrt[2*(8 + 3^(1/3) - I*3^(5/6))]])/    
   > > (3*Sqrt[(1/2)*(8 + 3^(1/3) - I*3^(5/6))]) +    
   > > (2*I*ArcTanh[(3^(1/6)*(I + Sqrt[3]) - 4*I*x)/    
   > > Sqrt[2*(8 + 3^(1/3) + I*3^(5/6))]])/    
   > > (3*Sqrt[(1/2)*(8 + 3^(1/3) + I*3^(5/6))])    
   > >   
   > Derive 6.10 handles this in one step by one rule:    
   >    
   > " INT(F'(x)/F(x)^n,x) -> -1/((n-1)*F(x)^(n-1)) "    
   >    
   > Perhaps Rubi should be taught this rule as well?    
   >    
   > Martin.   
      
   Here's a similar example where Hermite reduction should return the integral   
   without any logarithmic terms:     
      
   In[1793]:= (x^3 (2 + x^2))/(-2 + Sqrt[3] + (-2 + Sqrt[3]) x^2 - x^4)^2;   
   Integrate[%, x]   
   D[%, x] - %% // Simplify   
      
   Out[1794]= (((-2 - I) + Sqrt[3] - 2 x^2) ((1 + 2 I) - I Sqrt[3] +    
        2 I x^2) (x^4 (-2 I + ((2 - I) + Sqrt[3]) x^2) +    
        2 (-52 +    
           30 Sqrt[3] + ((-78 - 7 I) + (45 + 4 I) Sqrt[   
               3]) x^2 + ((-40 - 7 I) + (23 + 4 I) Sqrt[   
               3]) x^4 + ((-7 - 2 I) + (4 + I) Sqrt[3]) x^6) Log[(   
          2 (-2 + Sqrt[3]) + ((-2 - I) + Sqrt[3]) x^2)/(   
          2 (-2 + Sqrt[3]) + ((-2 + I) + Sqrt[3]) x^2)] +    
        2 (-52 +    
           30 Sqrt[3] + ((-78 - 7 I) + (45 + 4 I) Sqrt[   
               3]) x^2 + ((-40 - 7 I) + (23 + 4 I) Sqrt[   
               3]) x^4 + ((-7 - 2 I) + (4 + I) Sqrt[3]) x^6) Log[(   
          2 (-2 + Sqrt[3]) + ((-2 + I) + Sqrt[3]) x^2)/(   
          2 (-2 + Sqrt[3]) + ((-2 - I) + Sqrt[3]) x^2)]))/(8 (2 (-2 +    
           Sqrt[3]) + ((-2 - I) + Sqrt[3]) x^2) (-2 + Sqrt[   
        3] + (-2 + Sqrt[3]) x^2 - x^4)^2)   
      
   Out[1795]= 0   
      
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