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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
cos(x)     x      x*sin(x)          
------ + ------ - -------- + atan(x)
  2           2      2              
         1 + x
(−xsin⁡(x)2+(xx2+1+cos⁡(x)2))+atan⁡(x)\left(- \frac{x \sin{\left(x \right)}}{2} + \left(\frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2}\right)\right) + \operatorname{atan}{\left(x \right)} 
cos(x)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Simplificación general [src] 
    /     2\                                
    \1 + x /*(2*atan(x) - x*sin(x) + cos(x))
x + ----------------------------------------
                       2                    
--------------------------------------------
                        2                   
                   1 + x
x+(x2+1)(−xsin⁡(x)+cos⁡(x)+2atan⁡(x))2x2+1\frac{x + \frac{\left(x^{2} + 1\right) \left(- x \sin{\left(x \right)} + \cos{\left(x \right)} + 2 \operatorname{atan}{\left(x \right)}\right)}{2}}{x^{2} + 1} 
(x + (1 + x^2)*(2*atan(x) - x*sin(x) + cos(x))/2)/(1 + x^2)
Respuesta numérica [src] 
0.5*cos(x) + x/(1.0 + x^2) - 0.5*x*sin(x) + atan(x)
0.5*cos(x) + x/(1.0 + x^2) - 0.5*x*sin(x) + atan(x)
Denominador racional [src] 
      /       2\             /     2\            /       2\       
4*x + \4 + 4*x /*atan(x) + 2*\1 + x /*cos(x) - x*\2 + 2*x /*sin(x)
------------------------------------------------------------------
                                    2                             
                             4 + 4*x
−x(2x2+2)sin⁡(x)+4x+2(x2+1)cos⁡(x)+(4x2+4)atan⁡(x)4x2+4\frac{- x \left(2 x^{2} + 2\right) \sin{\left(x \right)} + 4 x + 2 \left(x^{2} + 1\right) \cos{\left(x \right)} + \left(4 x^{2} + 4\right) \operatorname{atan}{\left(x \right)}}{4 x^{2} + 4} 
(4*x + (4 + 4*x^2)*atan(x) + 2*(1 + x^2)*cos(x) - x*(2 + 2*x^2)*sin(x))/(4 + 4*x^2)
Abrimos la expresión [src] 
cos(x)     x      x*sin(x)          
------ + ------ - -------- + atan(x)
  2           2      2              
         1 + x
−xsin⁡(x)2+xx2+1+cos⁡(x)2+atan⁡(x)- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)} 
cos(x)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Potencias [src] 
 I*x    -I*x                /   -I*x    I*x\          
e      e         x      I*x*\- e     + e   /          
---- + ----- + ------ + -------------------- + atan(x)
 4       4          2            4                    
               1 + x
ix(eix−e−ix)4+xx2+1+eix4+atan⁡(x)+e−ix4\frac{i x \left(e^{i x} - e^{- i x}\right)}{4} + \frac{x}{x^{2} + 1} + \frac{e^{i x}}{4} + \operatorname{atan}{\left(x \right)} + \frac{e^{- i x}}{4} 
cos(x)     x      x*sin(x)          
------ + ------ - -------- + atan(x)
  2           2      2              
         1 + x
−xsin⁡(x)2+xx2+1+cos⁡(x)2+atan⁡(x)- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)} 
cos(x)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Compilar la expresión [src] 
cos(x)     x      x*sin(x)          
------ + ------ - -------- + atan(x)
  2           2      2              
         1 + x
−xsin⁡(x)2+xx2+1+cos⁡(x)2+atan⁡(x)- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)} 
cos(x)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Parte trigonométrica [src] 
   1         x            x                
-------- + ------ - ------------- + atan(x)
2*sec(x)        2        /    pi\          
           1 + x    2*sec|x - --|          
                         \    2 /
−x2sec⁡(x−π2)+xx2+1+atan⁡(x)+12sec⁡(x)- \frac{x}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} + \frac{1}{2 \sec{\left(x \right)}} 
                       /    pi\          
                  x*cos|x - --|          
cos(x)     x           \    2 /          
------ + ------ - ------------- + atan(x)
  2           2         2                
         1 + x
−xcos⁡(x−π2)2+xx2+1+cos⁡(x)2+atan⁡(x)- \frac{x \cos{\left(x - \frac{\pi}{2} \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)} 
   1         x         x              
-------- + ------ - -------- + atan(x)
2*sec(x)        2   2*csc(x)          
           1 + x
−x2csc⁡(x)+xx2+1+atan⁡(x)+12sec⁡(x)- \frac{x}{2 \csc{\left(x \right)}} + \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} + \frac{1}{2 \sec{\left(x \right)}} 
                  2/x\            /x\           
           1 - tan |-|       x*tan|-|           
  x                \2/            \2/           
------ + --------------- - ----------- + atan(x)
     2     /       2/x\\          2/x\          
1 + x    2*|1 + tan |-||   1 + tan |-|          
           \        \2//           \2/
−xtan⁡(x2)tan⁡2(x2)+1+xx2+1+1−tan⁡2(x2)2(tan⁡2(x2)+1)+atan⁡(x)- \frac{x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{x}{x^{2} + 1} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \operatorname{atan}{\left(x \right)} 
                   2/x\           /x\           
           -1 + cot |-|      x*cot|-|           
  x                 \2/           \2/           
------ + --------------- - ----------- + atan(x)
     2     /       2/x\\          2/x\          
1 + x    2*|1 + cot |-||   1 + cot |-|          
           \        \2//           \2/
−xcot⁡(x2)cot⁡2(x2)+1+xx2+1+cot⁡2(x2)−12(cot⁡2(x2)+1)+atan⁡(x)- \frac{x \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{x}{x^{2} + 1} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \operatorname{atan}{\left(x \right)} 
cos(x)     x      x*sin(x)          
------ + ------ - -------- + atan(x)
  2           2      2              
         1 + x
−xsin⁡(x)2+xx2+1+cos⁡(x)2+atan⁡(x)- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)} 
      1           x         x              
------------- + ------ - -------- + atan(x)
     /pi    \        2   2*csc(x)          
2*csc|-- - x|   1 + x                      
     \2     /
−x2csc⁡(x)+xx2+1+atan⁡(x)+12csc⁡(−x+π2)- \frac{x}{2 \csc{\left(x \right)}} + \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} + \frac{1}{2 \csc{\left(- x + \frac{\pi}{2} \right)}} 
   /    pi\                              
sin|x + --|                              
   \    2 /     x      x*sin(x)          
----------- + ------ - -------- + atan(x)
     2             2      2              
              1 + x
−xsin⁡(x)2+xx2+1+sin⁡(x+π2)2+atan⁡(x)- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{2} + \operatorname{atan}{\left(x \right)} 
sin(x + pi/2)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Denominador común [src] 
cos(x)     x      x*sin(x)          
------ + ------ - -------- + atan(x)
  2           2      2              
         1 + x
−xsin⁡(x)2+xx2+1+cos⁡(x)2+atan⁡(x)- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)} 
cos(x)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Unión de expresiones racionales [src] 
      /     2\            /     2\             /     2\       
2*x + \1 + x /*cos(x) + 2*\1 + x /*atan(x) - x*\1 + x /*sin(x)
--------------------------------------------------------------
                            /     2\                          
                          2*\1 + x /
−x(x2+1)sin⁡(x)+2x+(x2+1)cos⁡(x)+2(x2+1)atan⁡(x)2(x2+1)\frac{- x \left(x^{2} + 1\right) \sin{\left(x \right)} + 2 x + \left(x^{2} + 1\right) \cos{\left(x \right)} + 2 \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}{2 \left(x^{2} + 1\right)} 
(2*x + (1 + x^2)*cos(x) + 2*(1 + x^2)*atan(x) - x*(1 + x^2)*sin(x))/(2*(1 + x^2))
Combinatoria [src] 
 /                                        3           2             2        \ 
-\-cos(x) - 2*x - 2*atan(x) + x*sin(x) + x *sin(x) - x *cos(x) - 2*x *atan(x)/ 
-------------------------------------------------------------------------------
                                     /     2\                                  
                                   2*\1 + x /
−x3sin⁡(x)−x2cos⁡(x)−2x2atan⁡(x)+xsin⁡(x)−2x−cos⁡(x)−2atan⁡(x)2(x2+1)- \frac{x^{3} \sin{\left(x \right)} - x^{2} \cos{\left(x \right)} - 2 x^{2} \operatorname{atan}{\left(x \right)} + x \sin{\left(x \right)} - 2 x - \cos{\left(x \right)} - 2 \operatorname{atan}{\left(x \right)}}{2 \left(x^{2} + 1\right)} 
-(-cos(x) - 2*x - 2*atan(x) + x*sin(x) + x^3*sin(x) - x^2*cos(x) - 2*x^2*atan(x))/(2*(1 + x^2))
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