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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   cos(x)     (-2*x + sin(x))*sin(x)
----------- + ----------------------
 2                             2    
x  + cos(x)       / 2         \     
                  \x  + cos(x)/
(−2x+sin⁡(x))sin⁡(x)(x2+cos⁡(x))2+cos⁡(x)x2+cos⁡(x)\frac{\left(- 2 x + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\left(x^{2} + \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} + \cos{\left(x \right)}} 
cos(x)/(x^2 + cos(x)) + ((-2*x + sin(x))*sin(x))/(x^2 + cos(x))^2
Simplificación general [src] 
     2                    
1 + x *cos(x) - 2*x*sin(x)
--------------------------
                   2      
      / 2         \       
      \x  + cos(x)/
x2cos⁡(x)−2xsin⁡(x)+1(x2+cos⁡(x))2\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)} + 1}{\left(x^{2} + \cos{\left(x \right)}\right)^{2}} 
(1 + x^2*cos(x) - 2*x*sin(x))/(x^2 + cos(x))^2
Respuesta numérica [src] 
cos(x)/(x^2 + cos(x)) + (-2.0*x + sin(x))*sin(x)/(x^2 + cos(x))^2
cos(x)/(x^2 + cos(x)) + (-2.0*x + sin(x))*sin(x)/(x^2 + cos(x))^2
Combinatoria [src] 
   2         2       2                    
cos (x) + sin (x) + x *cos(x) - 2*x*sin(x)
------------------------------------------
                           2              
              / 2         \               
              \x  + cos(x)/
x2cos⁡(x)−2xsin⁡(x)+sin⁡2(x)+cos⁡2(x)(x2+cos⁡(x))2\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\left(x^{2} + \cos{\left(x \right)}\right)^{2}} 
(cos(x)^2 + sin(x)^2 + x^2*cos(x) - 2*x*sin(x))/(x^2 + cos(x))^2
Denominador racional [src] 
             2                                              
/ 2         \           / 2         \                       
\x  + cos(x)/ *cos(x) + \x  + cos(x)/*(-2*x + sin(x))*sin(x)
------------------------------------------------------------
                                    3                       
                       / 2         \                        
                       \x  + cos(x)/
(−2x+sin⁡(x))(x2+cos⁡(x))sin⁡(x)+(x2+cos⁡(x))2cos⁡(x)(x2+cos⁡(x))3\frac{\left(- 2 x + \sin{\left(x \right)}\right) \left(x^{2} + \cos{\left(x \right)}\right) \sin{\left(x \right)} + \left(x^{2} + \cos{\left(x \right)}\right)^{2} \cos{\left(x \right)}}{\left(x^{2} + \cos{\left(x \right)}\right)^{3}} 
((x^2 + cos(x))^2*cos(x) + (x^2 + cos(x))*(-2*x + sin(x))*sin(x))/(x^2 + cos(x))^3
Potencias [src] 
    I*x    -I*x                        /         /   -I*x    I*x\\
   e      e           /   -I*x    I*x\ |       I*\- e     + e   /|
   ---- + -----     I*\- e     + e   /*|-2*x - ------------------|
    2       2                          \               2         /
----------------- - ----------------------------------------------
      I*x    -I*x                                    2            
 2   e      e                     /      I*x    -I*x\             
x  + ---- + -----                 | 2   e      e    |             
      2       2                 2*|x  + ---- + -----|             
                                  \      2       2  /
−i(−2x−i(eix−e−ix)2)(eix−e−ix)2(x2+eix2+e−ix2)2+eix2+e−ix2x2+eix2+e−ix2- \frac{i \left(- 2 x - \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right) \left(e^{i x} - e^{- i x}\right)}{2 \left(x^{2} + \frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{x^{2} + \frac{e^{i x}}{2} + \frac{e^{- i x}}{2}} 
(exp(i*x)/2 + exp(-i*x)/2)/(x^2 + exp(i*x)/2 + exp(-i*x)/2) - i*(-exp(-i*x) + exp(i*x))*(-2*x - i*(-exp(-i*x) + exp(i*x))/2)/(2*(x^2 + exp(i*x)/2 + exp(-i*x)/2)^2)
Abrimos la expresión [src] 
            2                                                        
         sin (x)                cos(x)             2*x*sin(x)        
-------------------------- + ----------- - --------------------------
 4      2         2           2             4      2         2       
x  + cos (x) + 2*x *cos(x)   x  + cos(x)   x  + cos (x) + 2*x *cos(x)
−2xsin⁡(x)x4+2x2cos⁡(x)+cos⁡2(x)+sin⁡2(x)x4+2x2cos⁡(x)+cos⁡2(x)+cos⁡(x)x2+cos⁡(x)- \frac{2 x \sin{\left(x \right)}}{x^{4} + 2 x^{2} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)}}{x^{4} + 2 x^{2} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{x^{2} + \cos{\left(x \right)}} 
sin(x)^2/(x^4 + cos(x)^2 + 2*x^2*cos(x)) + cos(x)/(x^2 + cos(x)) - 2*x*sin(x)/(x^4 + cos(x)^2 + 2*x^2*cos(x))
Unión de expresiones racionales [src] 
/ 2         \                                
\x  + cos(x)/*cos(x) + (-2*x + sin(x))*sin(x)
---------------------------------------------
                             2               
                / 2         \                
                \x  + cos(x)/
(−2x+sin⁡(x))sin⁡(x)+(x2+cos⁡(x))cos⁡(x)(x2+cos⁡(x))2\frac{\left(- 2 x + \sin{\left(x \right)}\right) \sin{\left(x \right)} + \left(x^{2} + \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\left(x^{2} + \cos{\left(x \right)}\right)^{2}} 
((x^2 + cos(x))*cos(x) + (-2*x + sin(x))*sin(x))/(x^2 + cos(x))^2
Parte trigonométrica [src] 
          2             
         x         2*x  
1 + ----------- - ------
       /pi    \   csc(x)
    csc|-- - x|         
       \2     /         
------------------------
                    2   
  / 2        1     \    
  |x  + -----------|    
  |        /pi    \|    
  |     csc|-- - x||    
  \        \2     //
x2csc⁡(−x+π2)−2xcsc⁡(x)+1(x2+1csc⁡(−x+π2))2\frac{\frac{x^{2}}{\csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{2 x}{\csc{\left(x \right)}} + 1}{\left(x^{2} + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2}} 
                                       /              /x\ \         
                                       |         2*tan|-| |         
                                       |              \2/ |    /x\  
                                     2*|-2*x + -----------|*tan|-|  
                 2/x\                  |              2/x\|    \2/  
          1 - tan |-|                  |       1 + tan |-||         
                  \2/                  \               \2//         
-------------------------------- + ---------------------------------
              /            2/x\\                                   2
              |     1 - tan |-||                 /            2/x\\ 
/       2/x\\ | 2           \2/|                 |     1 - tan |-|| 
|1 + tan |-||*|x  + -----------|   /       2/x\\ | 2           \2/| 
\        \2// |            2/x\|   |1 + tan |-||*|x  + -----------| 
              |     1 + tan |-||   \        \2// |            2/x\| 
              \             \2//                 |     1 + tan |-|| 
                                                 \             \2//
1−tan⁡2(x2)(x2+1−tan⁡2(x2)tan⁡2(x2)+1)(tan⁡2(x2)+1)+2(−2x+2tan⁡(x2)tan⁡2(x2)+1)tan⁡(x2)(x2+1−tan⁡2(x2)tan⁡2(x2)+1)2(tan⁡2(x2)+1)\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(x^{2} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{2 \left(- 2 x + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \tan{\left(\frac{x}{2} \right)}}{\left(x^{2} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                                          1                
                                        ------ - 2*x       
              1                         csc(x)             
------------------------------ + --------------------------
/ 2        1     \    /pi    \                     2       
|x  + -----------|*csc|-- - x|   / 2        1     \        
|        /pi    \|    \2     /   |x  + -----------| *csc(x)
|     csc|-- - x||               |        /pi    \|        
\        \2     //               |     csc|-- - x||        
                                 \        \2     //
−2x+1csc⁡(x)(x2+1csc⁡(−x+π2))2csc⁡(x)+1(x2+1csc⁡(−x+π2))csc⁡(−x+π2)\frac{- 2 x + \frac{1}{\csc{\left(x \right)}}}{\left(x^{2} + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(x^{2} + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}} 
     /    pi\                            
  sin|x + --|                            
     \    2 /      (-2*x + sin(x))*sin(x)
---------------- + ----------------------
 2      /    pi\                      2  
x  + sin|x + --|    / 2      /    pi\\   
        \    2 /    |x  + sin|x + --||   
                    \        \    2 //
(−2x+sin⁡(x))sin⁡(x)(x2+sin⁡(x+π2))2+sin⁡(x+π2)x2+sin⁡(x+π2)\frac{\left(- 2 x + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\left(x^{2} + \sin{\left(x + \frac{\pi}{2} \right)}\right)^{2}} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{x^{2} + \sin{\left(x + \frac{\pi}{2} \right)}} 
                                        /              /x\ \          
                                        |         2*cot|-| |          
                                        |              \2/ |    /x\   
                                      2*|-2*x + -----------|*cot|-|   
                   2/x\                 |              2/x\|    \2/   
           -1 + cot |-|                 |       1 + cot |-||          
                    \2/                 \               \2//          
--------------------------------- + ----------------------------------
              /             2/x\\                                    2
              |     -1 + cot |-||                 /             2/x\\ 
/       2/x\\ | 2            \2/|                 |     -1 + cot |-|| 
|1 + cot |-||*|x  + ------------|   /       2/x\\ | 2            \2/| 
\        \2// |            2/x\ |   |1 + cot |-||*|x  + ------------| 
              |     1 + cot |-| |   \        \2// |            2/x\ | 
              \             \2/ /                 |     1 + cot |-| | 
                                                  \             \2/ /
2(−2x+2cot⁡(x2)cot⁡2(x2)+1)cot⁡(x2)(x2+cot⁡2(x2)−1cot⁡2(x2)+1)2(cot⁡2(x2)+1)+cot⁡2(x2)−1(x2+cot⁡2(x2)−1cot⁡2(x2)+1)(cot⁡2(x2)+1)\frac{2 \left(- 2 x + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \cot{\left(\frac{x}{2} \right)}}{\left(x^{2} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(x^{2} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
       2                
      x          2*x    
1 + ------ - -----------
    sec(x)      /    pi\
             sec|x - --|
                \    2 /
------------------------
                  2     
     / 2     1   \      
     |x  + ------|      
     \     sec(x)/
x2sec⁡(x)−2xsec⁡(x−π2)+1(x2+1sec⁡(x))2\frac{\frac{x^{2}}{\sec{\left(x \right)}} - \frac{2 x}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1}{\left(x^{2} + \frac{1}{\sec{\left(x \right)}}\right)^{2}} 
     2 /        2/x\\           /x\
    x *|-1 + cot |-||    4*x*cot|-|
       \         \2//           \2/
1 + ----------------- - -----------
              2/x\             2/x\
       1 + cot |-|      1 + cot |-|
               \2/              \2/
-----------------------------------
                           2       
        /             2/x\\        
        |     -1 + cot |-||        
        | 2            \2/|        
        |x  + ------------|        
        |            2/x\ |        
        |     1 + cot |-| |        
        \             \2/ /
x2(cot⁡2(x2)−1)cot⁡2(x2)+1−4xcot⁡(x2)cot⁡2(x2)+1+1(x2+cot⁡2(x2)−1cot⁡2(x2)+1)2\frac{\frac{x^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 x \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1}{\left(x^{2} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
     2 /       2/x\\           /x\
    x *|1 - tan |-||    4*x*tan|-|
       \        \2//           \2/
1 + ---------------- - -----------
             2/x\             2/x\
      1 + tan |-|      1 + tan |-|
              \2/              \2/
----------------------------------
                         2        
       /            2/x\\         
       |     1 - tan |-||         
       | 2           \2/|         
       |x  + -----------|         
       |            2/x\|         
       |     1 + tan |-||         
       \             \2//
x2(1−tan⁡2(x2))tan⁡2(x2)+1−4xtan⁡(x2)tan⁡2(x2)+1+1(x2+1−tan⁡2(x2)tan⁡2(x2)+1)2\frac{\frac{x^{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{4 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1}{\left(x^{2} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
     2                    
1 + x *cos(x) - 2*x*sin(x)
--------------------------
                   2      
      / 2         \       
      \x  + cos(x)/
x2cos⁡(x)−2xsin⁡(x)+1(x2+cos⁡(x))2\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)} + 1}{\left(x^{2} + \cos{\left(x \right)}\right)^{2}} 
                                1                
                           ----------- - 2*x     
                              /    pi\           
                           sec|x - --|           
         1                    \    2 /           
-------------------- + --------------------------
/ 2     1   \                       2            
|x  + ------|*sec(x)   / 2     1   \     /    pi\
\     sec(x)/          |x  + ------| *sec|x - --|
                       \     sec(x)/     \    2 /
−2x+1sec⁡(x−π2)(x2+1sec⁡(x))2sec⁡(x−π2)+1(x2+1sec⁡(x))sec⁡(x)\frac{- 2 x + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}}{\left(x^{2} + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(x^{2} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}} 
              /          /    pi\\    /    pi\
              |-2*x + cos|x - --||*cos|x - --|
   cos(x)     \          \    2 //    \    2 /
----------- + --------------------------------
 2                                  2         
x  + cos(x)            / 2         \          
                       \x  + cos(x)/
(−2x+cos⁡(x−π2))cos⁡(x−π2)(x2+cos⁡(x))2+cos⁡(x)x2+cos⁡(x)\frac{\left(- 2 x + \cos{\left(x - \frac{\pi}{2} \right)}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(x^{2} + \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{x^{2} + \cos{\left(x \right)}} 
     2    /    pi\             
1 + x *sin|x + --| - 2*x*sin(x)
          \    2 /             
-------------------------------
                        2      
      / 2      /    pi\\       
      |x  + sin|x + --||       
      \        \    2 //
x2sin⁡(x+π2)−2xsin⁡(x)+1(x2+sin⁡(x+π2))2\frac{x^{2} \sin{\left(x + \frac{\pi}{2} \right)} - 2 x \sin{\left(x \right)} + 1}{\left(x^{2} + \sin{\left(x + \frac{\pi}{2} \right)}\right)^{2}} 
                              1             
                            ------ - 2*x    
         1                  csc(x)          
-------------------- + ---------------------
/ 2     1   \                       2       
|x  + ------|*sec(x)   / 2     1   \        
\     sec(x)/          |x  + ------| *csc(x)
                       \     sec(x)/
−2x+1csc⁡(x)(x2+1sec⁡(x))2csc⁡(x)+1(x2+1sec⁡(x))sec⁡(x)\frac{- 2 x + \frac{1}{\csc{\left(x \right)}}}{\left(x^{2} + \frac{1}{\sec{\left(x \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(x^{2} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}} 
     2                 /    pi\
1 + x *cos(x) - 2*x*cos|x - --|
                       \    2 /
-------------------------------
                      2        
         / 2         \         
         \x  + cos(x)/
x2cos⁡(x)−2xcos⁡(x−π2)+1(x2+cos⁡(x))2\frac{x^{2} \cos{\left(x \right)} - 2 x \cos{\left(x - \frac{\pi}{2} \right)} + 1}{\left(x^{2} + \cos{\left(x \right)}\right)^{2}} 
(1 + x^2*cos(x) - 2*x*cos(x - pi/2))/(x^2 + cos(x))^2
Denominador común [src] 
   2         2       2                    
cos (x) + sin (x) + x *cos(x) - 2*x*sin(x)
------------------------------------------
         4      2         2               
        x  + cos (x) + 2*x *cos(x)
x2cos⁡(x)−2xsin⁡(x)+sin⁡2(x)+cos⁡2(x)x4+2x2cos⁡(x)+cos⁡2(x)\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{x^{4} + 2 x^{2} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} 
(cos(x)^2 + sin(x)^2 + x^2*cos(x) - 2*x*sin(x))/(x^4 + cos(x)^2 + 2*x^2*cos(x))
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