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How do you (a^-2-a/a^-1-1)/(a^2+1) in partial fractions?

An expression to simplify:

The solution

You have entered [src]
1     a     
-- - --- - 1
 2   /1\    
a    |-|    
     \a/    
------------
    2       
   a  + 1   
$$\frac{\left(\frac{1}{a^{2}} - \frac{a}{\frac{1}{a}}\right) - 1}{a^{2} + 1}$$
(a^(-2) - a/1/a - 1)/(a^2 + 1)
Fraction decomposition [src]
-1 + a^(-2) - 1/(1 + a^2)
$$-1 - \frac{1}{a^{2} + 1} + \frac{1}{a^{2}}$$
     1      1   
-1 + -- - ------
      2        2
     a    1 + a 
General simplification [src]
     2    4
1 - a  - a 
-----------
 2 /     2\
a *\1 + a /
$$\frac{- a^{4} - a^{2} + 1}{a^{2} \left(a^{2} + 1\right)}$$
(1 - a^2 - a^4)/(a^2*(1 + a^2))
Rational denominator [src]
     2    4
1 - a  - a 
-----------
 2 /     2\
a *\1 + a /
$$\frac{- a^{4} - a^{2} + 1}{a^{2} \left(a^{2} + 1\right)}$$
(1 - a^2 - a^4)/(a^2*(1 + a^2))
Numerical answer [src]
(-1.0 + a^(-2) - a^2)/(1.0 + a^2)
(-1.0 + a^(-2) - a^2)/(1.0 + a^2)
Combining rational expressions [src]
     2    4
1 - a  - a 
-----------
 2 /     2\
a *\1 + a /
$$\frac{- a^{4} - a^{2} + 1}{a^{2} \left(a^{2} + 1\right)}$$
(1 - a^2 - a^4)/(a^2*(1 + a^2))
Combinatorics [src]
 /      2    4\ 
-\-1 + a  + a / 
----------------
   2 /     2\   
  a *\1 + a /   
$$- \frac{a^{4} + a^{2} - 1}{a^{2} \left(a^{2} + 1\right)}$$
-(-1 + a^2 + a^4)/(a^2*(1 + a^2))
Powers [src]
     1     2
-1 + -- - a 
      2     
     a      
------------
        2   
   1 + a    
$$\frac{- a^{2} - 1 + \frac{1}{a^{2}}}{a^{2} + 1}$$
(-1 + a^(-2) - a^2)/(1 + a^2)
Common denominator [src]
        1   
-1 + -------
      2    4
     a  + a 
$$-1 + \frac{1}{a^{4} + a^{2}}$$
-1 + 1/(a^2 + a^4)
Trigonometric part [src]
     1     2
-1 + -- - a 
      2     
     a      
------------
        2   
   1 + a    
$$\frac{- a^{2} - 1 + \frac{1}{a^{2}}}{a^{2} + 1}$$
(-1 + a^(-2) - a^2)/(1 + a^2)
Assemble expression [src]
     1     2
-1 + -- - a 
      2     
     a      
------------
        2   
   1 + a    
$$\frac{- a^{2} - 1 + \frac{1}{a^{2}}}{a^{2} + 1}$$
(-1 + a^(-2) - a^2)/(1 + a^2)
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