integrate(1/(sin(x)^2+1),x,a,b)?



> From: Raymond Toy
> Subject: integrate(1/(sin(x)^2+1),x,a,b)?
> 
> What should maxima do here?  What do Macsyma, Maple, and/or
> Mathematica return for this integral?
> 
> Ray

Mathematica 6.0 for Linux x86 (64-bit)
Copyright 1988-2007 Wolfram Research, Inc.

In[2]:=  Integrate[1/(Sin[x]^2+1),{x,0,z}]
 
                          -Pi             Pi
Out[2]= If[(Im[z] != 0 || --- <= Re[z] <= --) &&
                           2              2
  
>     (ArcCosh[3] + 2 Im[z] >= 0 || z == Re[z] || Re[z] != 0),
  
     ArcTan[Sqrt[2] Tan[z]]                  z
>    ----------------------, Integrate[-------------, {x, 0, 1},
            Sqrt[2]                                2
                                       1 + Sin[x z]
  
>     Assumptions ->
  
                        -Pi             Pi
>      !((Im[z] != 0 || --- <= Re[z] <= --) &&
                         2              2
  
                    I                                      I
>         (Im[z] == - ArcCos[3] || (Im[z] <= 0 && Im[z] >= - ArcCos[3]) ||
                    2                                      2
  
>           Im[z] == 0 || Im[z] >= 0 || Re[z] != 0))]]


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