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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
            x        1     atan(x)
atan(x) + ------ - ----- - -------
               2   x + 1     2    
          1 + x             x  + 1
((xx2+1+atan⁡(x))−1x+1)−atan⁡(x)x2+1\left(\left(\frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)}\right) - \frac{1}{x + 1}\right) - \frac{\operatorname{atan}{\left(x \right)}}{x^{2} + 1} 
atan(x) + x/(1 + x^2) - 1/(x + 1) - atan(x)/(x^2 + 1)
Simplificación general [src] 
          2            3        
-1 + x + x *atan(x) + x *atan(x)
--------------------------------
                 2    3         
        1 + x + x  + x
x3atan⁡(x)+x2atan⁡(x)+x−1x3+x2+x+1\frac{x^{3} \operatorname{atan}{\left(x \right)} + x^{2} \operatorname{atan}{\left(x \right)} + x - 1}{x^{3} + x^{2} + x + 1} 
(-1 + x + x^2*atan(x) + x^3*atan(x))/(1 + x + x^2 + x^3)
Respuesta numérica [src] 
-1/(1.0 + x) + x/(1.0 + x^2) - atan(x)/(1.0 + x^2) + atan(x)
-1/(1.0 + x) + x/(1.0 + x^2) - atan(x)/(1.0 + x^2) + atan(x)
Denominador racional [src] 
/     2\ /      2           /    /     2\        \\           /     2\        
\1 + x /*\-1 - x  + (1 + x)*\x + \1 + x /*atan(x)// - (1 + x)*\1 + x /*atan(x)
------------------------------------------------------------------------------
                                              2                               
                                      /     2\                                
                              (1 + x)*\1 + x /
−(x+1)(x2+1)atan⁡(x)+(x2+1)(−x2+(x+1)(x+(x2+1)atan⁡(x))−1)(x+1)(x2+1)2\frac{- \left(x + 1\right) \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)} + \left(x^{2} + 1\right) \left(- x^{2} + \left(x + 1\right) \left(x + \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}\right) - 1\right)}{\left(x + 1\right) \left(x^{2} + 1\right)^{2}} 
((1 + x^2)*(-1 - x^2 + (1 + x)*(x + (1 + x^2)*atan(x))) - (1 + x)*(1 + x^2)*atan(x))/((1 + x)*(1 + x^2)^2)
Compilar la expresión [src] 
    1       x      atan(x)          
- ----- + ------ - ------- + atan(x)
  1 + x        2         2          
          1 + x     1 + x
xx2+1+atan⁡(x)−atan⁡(x)x2+1−1x+1\frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} - \frac{\operatorname{atan}{\left(x \right)}}{x^{2} + 1} - \frac{1}{x + 1} 
-1/(1 + x) + x/(1 + x^2) - atan(x)/(1 + x^2) + atan(x)
Unión de expresiones racionales [src] 
      2           /    /     2\        \                  
-1 - x  + (1 + x)*\x + \1 + x /*atan(x)/ - (1 + x)*atan(x)
----------------------------------------------------------
                             /     2\                     
                     (1 + x)*\1 + x /
−x2+(x+1)(x+(x2+1)atan⁡(x))−(x+1)atan⁡(x)−1(x+1)(x2+1)\frac{- x^{2} + \left(x + 1\right) \left(x + \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}\right) - \left(x + 1\right) \operatorname{atan}{\left(x \right)} - 1}{\left(x + 1\right) \left(x^{2} + 1\right)} 
(-1 - x^2 + (1 + x)*(x + (1 + x^2)*atan(x)) - (1 + x)*atan(x))/((1 + x)*(1 + x^2))
Combinatoria [src] 
          2            3        
-1 + x + x *atan(x) + x *atan(x)
--------------------------------
                /     2\        
        (1 + x)*\1 + x /
x3atan⁡(x)+x2atan⁡(x)+x−1(x+1)(x2+1)\frac{x^{3} \operatorname{atan}{\left(x \right)} + x^{2} \operatorname{atan}{\left(x \right)} + x - 1}{\left(x + 1\right) \left(x^{2} + 1\right)} 
(-1 + x + x^2*atan(x) + x^3*atan(x))/((1 + x)*(1 + x^2))
Potencias [src] 
    1       x      atan(x)          
- ----- + ------ - ------- + atan(x)
  1 + x        2         2          
          1 + x     1 + x
xx2+1+atan⁡(x)−atan⁡(x)x2+1−1x+1\frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} - \frac{\operatorname{atan}{\left(x \right)}}{x^{2} + 1} - \frac{1}{x + 1} 
-1/(1 + x) + x/(1 + x^2) - atan(x)/(1 + x^2) + atan(x)
Parte trigonométrica [src] 
    1       x      atan(x)          
- ----- + ------ - ------- + atan(x)
  1 + x        2         2          
          1 + x     1 + x
xx2+1+atan⁡(x)−atan⁡(x)x2+1−1x+1\frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} - \frac{\operatorname{atan}{\left(x \right)}}{x^{2} + 1} - \frac{1}{x + 1} 
-1/(1 + x) + x/(1 + x^2) - atan(x)/(1 + x^2) + atan(x)
Denominador común [src] 
  1 - x + x*atan(x) + atan(x)          
- --------------------------- + atan(x)
                 2    3                
        1 + x + x  + x
atan⁡(x)−xatan⁡(x)−x+atan⁡(x)+1x3+x2+x+1\operatorname{atan}{\left(x \right)} - \frac{x \operatorname{atan}{\left(x \right)} - x + \operatorname{atan}{\left(x \right)} + 1}{x^{3} + x^{2} + x + 1} 
-(1 - x + x*atan(x) + atan(x))/(1 + x + x^2 + x^3) + atan(x)
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