From: Albert_Rich@msn.com   
      
   On Monday, November 21, 2016 at 6:35:36 PM UTC-10, Nasser M. Abbasi wrote:   
   > It is interesting how different CAS outputs differ sometimes.   
   > I was trying to verify book result for int(sec(t),t) and this   
   > is what some CAS systems gave   
   >    
   > Mathematica:     ln(cos(t/2)+sin(t/2))-ln(cos(t/2)-sin(t/2))   
   > Rubi:            arctanh(sin(t))   
   > Fricas:          1/2 ( ln(1+sin(t)) - ln(1-sin(t)) )   
   > Maple & Maxima:  ln(sec(t)+tan(t))   
   >    
   > Text Book happens to give result shown by Maple&Maxima.   
   >    
   > Ofcourse all other results are also correct, I am sure,   
   > and these can all be converted to each others.   
   >    
   > Was Wondering if there is any preference, math-wise, to any one   
   > of the above results. From the Mathematica result, it seems   
   > to hint that it used the Tangent half-angle substitution,   
   > also called Weierstrass substitution.   
   >    
   > What does you CAS give to int(sec(t),t) if different from   
   > the above?   
   >    
   > --Nasser   
      
   Note that in addition to being the most compact, Rubi's antiderivative is   
   always real when t is real, unlike the ones returned by Mathematica, Maple and   
   Maxima. Also the other antiderivatives involve multiple instances of the   
   integration variable.  This    
   can result in catastrophic cancellation when these antiderivatives are   
   numerically evaluated.      
      
   However, if you still prefer an alternative antiderivative for sec(t), note   
   that Rubi is an open-source, modular system.  So, all you have to do is change   
   just one of Rubi's 6800+ integration rules to get the desired effect.    
   Specifically the rule   
      
       int(sec(c+d x),x) --> arctanh(sin(c+d x))/d   
      
   Albert   
      
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