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Simplificación de expresiones/ Descomposición en fracciones simples/ exp(x)/(x^2+1)-2*x*exp(x)/(x^2+1)^2

¿Cómo vas a descomponer esta exp(x)/(x^2+1)-2*x*exp(x)/(x^2+1)^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   x            x 
  e        2*x*e  
------ - ---------
 2               2
x  + 1   / 2    \ 
         \x  + 1/
−2xex(x2+1)2+exx2+1- \frac{2 x e^{x}}{\left(x^{2} + 1\right)^{2}} + \frac{e^{x}}{x^{2} + 1} 
exp(x)/(x^2 + 1) - (2*x)*exp(x)/(x^2 + 1)^2
Simplificación general [src] 
/     2      \  x
\1 + x  - 2*x/*e 
-----------------
            2    
    /     2\     
    \1 + x /
(x2−2x+1)ex(x2+1)2\frac{\left(x^{2} - 2 x + 1\right) e^{x}}{\left(x^{2} + 1\right)^{2}} 
(1 + x^2 - 2*x)*exp(x)/(1 + x^2)^2
Respuesta numérica [src] 
exp(x)/(1.0 + x^2) - 2.0*x*exp(x)/(1.0 + x^2)^2
exp(x)/(1.0 + x^2) - 2.0*x*exp(x)/(1.0 + x^2)^2
Denominador común [src] 
 2  x        x    x
x *e  - 2*x*e  + e 
-------------------
        4      2   
   1 + x  + 2*x
x2ex−2xex+exx4+2x2+1\frac{x^{2} e^{x} - 2 x e^{x} + e^{x}}{x^{4} + 2 x^{2} + 1} 
(x^2*exp(x) - 2*x*exp(x) + exp(x))/(1 + x^4 + 2*x^2)
Denominador racional [src] 
        2                     
/     2\   x       /     2\  x
\1 + x / *e  - 2*x*\1 + x /*e 
------------------------------
                  3           
          /     2\            
          \1 + x /
−2x(x2+1)ex+(x2+1)2ex(x2+1)3\frac{- 2 x \left(x^{2} + 1\right) e^{x} + \left(x^{2} + 1\right)^{2} e^{x}}{\left(x^{2} + 1\right)^{3}} 
((1 + x^2)^2*exp(x) - 2*x*(1 + x^2)*exp(x))/(1 + x^2)^3
Combinatoria [src] 
        2  x
(-1 + x) *e 
------------
         2  
 /     2\   
 \1 + x /
(x−1)2ex(x2+1)2\frac{\left(x - 1\right)^{2} e^{x}}{\left(x^{2} + 1\right)^{2}} 
(-1 + x)^2*exp(x)/(1 + x^2)^2
Unión de expresiones racionales [src] 
/     2      \  x
\1 + x  - 2*x/*e 
-----------------
            2    
    /     2\     
    \1 + x /
(x2−2x+1)ex(x2+1)2\frac{\left(x^{2} - 2 x + 1\right) e^{x}}{\left(x^{2} + 1\right)^{2}} 
(1 + x^2 - 2*x)*exp(x)/(1 + x^2)^2
Parte trigonométrica [src] 
        2                    
(-1 + x) *(cosh(x) + sinh(x))
-----------------------------
                  2          
          /     2\           
          \1 + x /
(x−1)2(sinh⁡(x)+cosh⁡(x))(x2+1)2\frac{\left(x - 1\right)^{2} \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{\left(x^{2} + 1\right)^{2}} 
cosh(x) + sinh(x)   2*x*(cosh(x) + sinh(x))
----------------- - -----------------------
           2                       2       
      1 + x                /     2\        
                           \1 + x /
−2x(sinh⁡(x)+cosh⁡(x))(x2+1)2+sinh⁡(x)+cosh⁡(x)x2+1- \frac{2 x \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{\left(x^{2} + 1\right)^{2}} + \frac{\sinh{\left(x \right)} + \cosh{\left(x \right)}}{x^{2} + 1} 
(cosh(x) + sinh(x))/(1 + x^2) - 2*x*(cosh(x) + sinh(x))/(1 + x^2)^2
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