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writting an expression


Hello.

I have the expression

In[193]:=
ee = 1/(1 + o^4);

and I want to write in the form

Out[194]=
(-Sqrt[2] + o)/(2*Sqrt[2]*(-1 + Sqrt[2]*o - o^2)) + (Sqrt[2] + o)/
(2*Sqrt[2]*(1 + Sqrt[2]*o + o^2))

Currently I can think something like

In[237]:=
1 + o^4 == (a + b*o + o^2)*(c + d*o + o^2) + O[o]^5
LogicalExpand[%]
({ToRules[#1]} & )[(Reduce[#1, {a, b, c, d}] & )[Reduce[% && a < b,
Reals]]]
Apart[1/((a + b*o + o^2)*(c + d*o + o^2) /. %[[1]])]

Out[237]=
1 + o^4 == SeriesData[o, 0, {a*c, b*c + a*d, a + c + b*d, b + d, 1},
0, 5, 1]

Out[238]=
-1 + a*c == 0 && b + d == 0 && b*c + a*d == 0 && a + c + b*d == 0

Out[239]=
{{a -> 1, b -> Sqrt[2], c -> 1, d -> -Sqrt[2]}}

Out[240]=
(-Sqrt[2] + o)/(2*Sqrt[2]*(-1 + Sqrt[2]*o - o^2)) + (Sqrt[2] + o)/
(2*Sqrt[2]*(1 + Sqrt[2]*o + o^2))

Any other ideas?

Thanks.
Dimitris



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