From: Albert_Rich@msn.com   
      
   On Tuesday, January 31, 2017 at 6:40:08 AM UTC-10, clicl...@freenet.de wrote:   
   > oldk1331@gmail.com schrieb:   
   > >   
   > > > >   
   > > > > FriCAS 1.3   
   > > > > ==========   
   > > > > integrate( sqrt(1 - x^4)/(1 + x^4), x)   
   > > > >   
   > > > > (log((2*x*((-1)*x^4+1)^(1/2)+((-1)*x^4+2*x^2+1))/(x^4+1)   
   +((-2)*atan((((-1)*   
   > > > >    x^4+1)^(1/2)+x)/x)+2*atan(x^2)))/4   
   > > >   
   > > > This is (i) real and (ii) elementary, but (iii) discontinuous at x=0.   
   > > > There are no other discontinuities.   
   > > >   
   > > > -   
   > > >   
   > > > Only Maple 2016.2 and FriCAS 1.3 return elementary answers, but both are   
   > > > discontinuous at x=0.   
   > > >   
   > > > More work is therefore needed to make the output of these integrators   
   > > > Davenport compatible.   
   > > >   
   > > > Martin.   
   > >   
   > > Hi Martin, FriCAS does an extra simplification, causes the result to be   
   > > discontinuous at x=0.  This is caused by the same bug you mentioned   
   > > in the previous thread.  I think there is a good enough fix, and the   
   > > result after patching this bug is continuous:   
   > >   
   > > (log((2*x*((-1)*x^4+1)^(1/2)+((-1)*x^4+2*x^2+1))/(x^4+1))+2*   
   tan((x^2*((-1)*   
   > >   x^4+1)^(1/2)+(x^3+x))/(((-1)*x^4+1)^(1/2)+((-1)*x^3+x))))/4   
   > >   
   >   
   > This can still be simplified considerably:   
   >   
   >   1/2*ATANH(x*(1 - x^2)/SQRT(1 - x^4))   
   >   + 1/2*ATAN(x*(1 + x^2)/SQRT(1 - x^4))   
   >   
   > Martin.   
      
   Generalizing the above for integrands of the form sqrt(a+b*x^4)/(c+d*x^4) when   
   a*d+b*c=0 yields the following antiderivative:   
      
   a/(2*c*(-a*b)^(1/4))*   
       arctan((-a*b)^(1/4)*x*(a+sqrt(-a*b)*x^2)/(a*sqrt(a+b*x^4))) +   
   a/(2*c*(-a*b)^(1/4))*   
       arctanh((-a*b)^(1/4)*x*(a-sqrt(-a*b)*x^2)/(a*sqrt(a+b*x^4)))   
      
   The next release of Rubi will use this rule when a*b<0. Any ideas for when   
   a*b>0?   
      
   Albert   
      
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