>Maple does
>
> > 1/(1+I);
>                                   1/2 - 1/2 I

Axiom does

 1/(1+%i)

       1
     ------
     1 + %i

which is of type Fraction Complex Integer, that is a fraction 
whose numerator and denominator are of type Complex(Integer).

You can ask Axiom to place the result in a different type:

 1/(1+%i)::Complex Fraction Integer

  1   1
  - - - %i
  2   2

and the type is Complex Fraction Integer, that is a Complex 
number whose real and imaginary components are Fraction(Integer).

The "simple form" depends on what is canonical for the type.

Tim

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