From: Albert_Rich@msn.com   
      
   On Tuesday, July 9, 2019 at 6:12:16 AM UTC-10, clicl...@freenet.de wrote:   
   >   
   > integrate((x - sqrt(a^2 + 1) - a)/   
   > ((x + sqrt(a^2 + 1) - a)*sqrt((x - a)*(x^2 + 1))), x)   
      
   Factoring out the piecewise-constant sqrt(x-a)*sqrt(x^2+1)/sqrt((x-a)*(x^2+1))   
   and making the substitution t=sqrt(x-a), transforms this into a    
   seudo-elliptic integrand of the form   
      
       (A+B*x^2)/((d+e*x^2)*sqrt(a+b*x^2+c*x^4))   
      
   where c*d^2-a*e^2=0 and B*d+A*e=0.  That Rubi knows how to integrate by   
   transforming it into an elementary integrand of the form 1/(C+D*t^2) using the   
   substitution t=x/sqrt(a+b*x^2+c*x^4).   
      
   Albert   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)