Albert Rich schrieb:   
   >   
   > On Wednesday, December 14, 2016 at 7:05:47 AM UTC-10, clicl...@freenet.de   
   wrote:   
   > >   
   > > [...] What is the most appropriate antiderivative should follow from   
   > > your choice of design principles.   
   > >   
   > > In fact, I myself seem to have introduced antiderivatives of the   
   > > ATANH(SIN(t)) type in my sci.math.symbolic post of Sun, 21 Apr 2013   
   > > 18:20:53 +0200 to the thread "An independent integration test   
   > > suite".   
   > >   
   > > This form of antiderivative is real on the entire real line just as   
   > > the integrand is. The alternative LN(TAN(pi/4 + t/2)) is almost as   
   > > simple, but its branch cuts in the complex plane are less disruptive   
   > > while its imaginary part along the real line jumps between zero and   
   > > pi.   
   > >   
   > > Many of the antiderivatives that exhibit logarithmic poles should   
   > > admit two or more alternative forms like this.   
   > >   
   >   
   > A design principle I value highly is the symmetry between the trig and   
   > hyperbolic functions.  For the antiderivatives of sec(x) and csc(x),   
   > Rubi currently returns   
   >   
   >     arctanh(sin(x))  and  -arctanh(cos(x))   
   >   
   > respectively.  For the antiderivatives of sech(x) and csch(x), it   
   > returns   
   >   
   >     arctan(sinh(x))  and  -arctanh(cosh(x))   
   >   
   > respectively.  All nicely symmetric, except for there being just one   
   > arctan and but three arctanhs...   
   >   
   > I guess if a Rubi user changes the antiderivative of sec(x) to   
   >   
   >     log(tan(x/2+pi/4)),   
   >   
   > he or she should also change the antiderivative of csc(x) to   
   >   
   >     log(tan(x/2)).   
   >   
   > And for symmetry, should he or she also change the antiderivatives of   
   > sech(x) and csch(x) to   
   >   
   >     i*log(tanh(x/2+pi*i/4))  and  log(tanh(x/2))   
   >   
   > respectively, where i is the imaginary unit?   
   >   
      
   Among potential integrator design principles, such formal symmetries   
   rank fairly low in my mind. But they may inform my intuition about   
   alternative forms of antiderivatives.   
      
   As your INT(SECH(x), x) = #i*LOG(TANH(x/2 + pi*#i/4)) is no composite   
   function of real analysis, it would be incomprehensible to most college   
   students freshly introduced to calculus. Indeed, Derive 6.10 returns   
   2*ATAN(#e^x) instead. Here too I suspect that the branch cuts of your   
   ATAN(SINH(x)) disrupt the complex plane more severely.   
      
   Requiring real antiderivatives to be valid composite functions of real   
   analysis is an important design principle in my view.   
      
   Martin.   
      
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