integrate(1/(1+x^4),x=%minusInfinity..%plusInfinity)

(1)

is obviously wrong. UPDATE: this problem is fixed in FriCAS.

integrate(1/(1+x^4),x)

(2)

is an antiderivative only for . The correct antiderivative, i.e., which is defined on all of can be obtained from the above expression by replacing with . UPDATE: this problem is also fixed in FriCAS, although the result could be simpler. For example, mathematica 5.2 gives:

  In[10]:= Integrate[1/(1+x^4), x]

  Out[10]= (-2 ArcTan[1 - Sqrt[2] x] + 2 ArcTan[1 + Sqrt[2] x] - 

                               2                         2
         Log[-1 + Sqrt[2] x - x ] + Log[1 + Sqrt[2] x + x ]) / (4 Sqrt[2])

However,

integrate((x^4+2*a*x^2+1)^-1, x=0..%plusInfinity, "noPole")

(3)

is wrong, also in FriCAS. However, the solution really depends on the value of a. Mathematica 5.2 gives:

  In[2]:= Integrate[(x^4+2*a*x^2+1)^-1, {x,0,Infinity}]

                                     2
  Out[2]= If[Im[Sqrt[-a - Sqrt[-1 + a ]]] > 0 && 

                              2
      Im[Sqrt[-a + Sqrt[-1 + a ]]] > 0, 

      -I                             2                        2
     (-- Pi) / (Sqrt[-a - Sqrt[-1 + a ]] Sqrt[-a + Sqrt[-1 + a ]] 
      2

                              2                          2
        (Sqrt[-a - Sqrt[-1 + a ]] + Sqrt[-a + Sqrt[-1 + a ]])), 

                      1
     Integrate[---------------, {x, 0, Infinity}, 
                        2    4
               1 + 2 a x  + x

      Assumptions -> 

                               2                                    2
       Im[Sqrt[-a - Sqrt[-1 + a ]]] <= 0 || Im[Sqrt[-a + Sqrt[-1 + a ]]] <= 0]

      ]

The definite version seems to be correct:

D(integrate((x^4+2*a*x^2+1)^-1, x=0..b, "noPole"), b)

(4)

The following also seems to be wrong:

ex := integrate(1/(a+x**4),x=0..%plusInfinity, "noPole")

   There are no library operations named ** 
      Use HyperDoc Browse or issue
                                 )what op **
      to learn if there is any operation containing " ** " in its name.

   Cannot find a definition or applicable library operation named ** 
      with argument type(s) 
                                 Variable(x)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
eval(ex, a=1)

(5)

compare with

integrate(1/(1+x**4),x=0..%plusInfinity, "noPole")

   There are no library operations named ** 
      Use HyperDoc Browse or issue
                                 )what op **
      to learn if there is any operation containing " ** " in its name.

   Cannot find a definition or applicable library operation named ** 
      with argument type(s) 
                                 Variable(x)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Name: #293 integrate (1/(1+x^4),x = %minusInfinity..%plusInfinity) => #293 integrate (1/(1+x^4)

http://fricas.svn.sourceforge.net/viewvc/fricas/trunk/src/algebra/gaussian.spad.pamphlet?r1=257&r2=358&view=patch

I'm not sure it solves the issue though.

Name: #293 integrate 1/(1+x)^4 => #293 integrate 1/(1+x^4)

Name: #293 integrate 1/(4x^2-1) => #293 integrate 1/(1+x^4)

Name: #293 integrate 1/(1+x^4) => #293 integrate 1x/(1+x^4)

Name: #293 integrate 1x/(1+x^4) => #293 integrate 1/(1+x^4)

Name: #293 integrate 1/(1+x^4) => #293 integrate 1/(1+x^2)