From: Albert_Rich@msn.com   
      
   On Sunday, January 16, 2022 at 10:38:19 PM UTC-10, nob...@nowhere.invalid   
   wrote:   
   > Albert Rich schrieb:   
   > >    
   > > On Saturday, January 15, 2022 at 2:15:27 AM UTC-10, nob...@nowhere.invalid   
   wrote:    
   > >    
   > > > Euler's elementary algebraic integral:    
   > > >    
   > > > INT((1 + x^2)^2/((1 - x^2)*(1 - 6*x^2 + x^4)^(3/4)), x)    
   > > >    
   > > > from paper E689 in the Euler Archive has the solution:    
   > > >    
   > > > - ATANH(2*x*(x^4 - 6*x^2 + 1)^(1/4)/(SQRT(x^4 - 6*x^2 + 1) + x^2 + 1))    
   > > > + ATAN(x/((x^4 - 6*x^2 + 1)^(1/4) + 1))    
   > > > - ATAN(((x^4 - 6*x^2 + 1)^(1/4) - 1)/x)    
   > > >    
   > > > which is real and continuous where the integrand is real and    
   > > > continuous. It appears that the awkward ATANH term cannot be    
   > > > decomposed into simpler ones with real arguments.    
   > > >    
   > >   
   > > What's wrong with combining the two arctan terms to get in    
   > > Mathematica notation:    
   > >    
   > > ArcTan[(1 + x^2 - Sqrt[1 - 6*x^2 + x^4])/(2*x*(1 - 6*x^2 + x^4)^(1/4))] -    
   > > ArcTanh[(2*x*(1 - 6*x^2 + x^4)^(1/4))/(1 + x^2 + Sqrt[1 - 6*x^2 + x^4])]    
   > >    
   > > which is also real and continuous where the integrand is real and    
   > > continuous.    
   > >   
   > Nothing really, only that ATANHs and ATANs broken down as far as    
   > possible typically exhibit fewer unwarranted discontinuities; for the    
   > integral at hand, however, both formulations agree on the real axis.    
   >    
   > In fact, Euler himself gives both at the end of §13 in his "Integratio    
   > Formulae Differentialis Maxime Irrationalis quam Tamen per Logarithmos    
   > et Arcus Circulares Expedire Licet" of 1777 (paper E689).    
   >    
   > But fusing two ATANHs or two ATANs is easier than recognizing how they    
   > can perhaps be split. So, returning two terms instead of one may be    
   > considered a service to the reader or user.    
   >    
   > Martin.   
      
   Yes, fusing two arctangents into one composite arctangent is easier than   
   splitting one arctangent into two simpler arctangents.  This seems analogous   
   to the fact that multiplying two expressions is easier than factoring one   
   expression into two simpler    
   expressions.   
      
   There are numerous techniques for factoring expressions (integers,   
   polynomials, etc).  Are there techniques for “factoring” a function f(x)   
   into two simpler functions g(x) and h(x) such that   
      
       f(x) = (g(x)+h(x)) / (1-g(x)*h(x))   
      
   where neither g(x) nor h(x) are constant wrt x?   
      
   Albert   
      
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