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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   cos(2*x)*x           
--------------- - cos(x)
cos(x) + sin(x)
$$\frac{x \cos{\left(2 x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} - \cos{\left(x \right)}$$ 
(cos(2*x)*x)/(cos(x) + sin(x)) - cos(x)
Respuesta numérica [src] 
-cos(x) + x*cos(2*x)/(cos(x) + sin(x))
-cos(x) + x*cos(2*x)/(cos(x) + sin(x))
Unión de expresiones racionales [src] 
x*cos(2*x) - (cos(x) + sin(x))*cos(x)
-------------------------------------
           cos(x) + sin(x)
$$\frac{x \cos{\left(2 x \right)} - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}$$ 
(x*cos(2*x) - (cos(x) + sin(x))*cos(x))/(cos(x) + sin(x))
Potencias [src] 
                          / -2*I*x    2*I*x\      
                          |e         e     |      
   I*x    -I*x          x*|------- + ------|      
  e      e                \   2        2   /      
- ---- - ----- + ---------------------------------
   2       2      I*x    -I*x     /   -I*x    I*x\
                 e      e       I*\- e     + e   /
                 ---- + ----- - ------------------
                  2       2             2
$$\frac{x \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right)}{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + \frac{e^{i x}}{2} + \frac{e^{- i x}}{2}} - \frac{e^{i x}}{2} - \frac{e^{- i x}}{2}$$ 
-exp(i*x)/2 - exp(-i*x)/2 + x*(exp(-2*i*x)/2 + exp(2*i*x)/2)/(exp(i*x)/2 + exp(-i*x)/2 - i*(-exp(-i*x) + exp(i*x))/2)
Combinatoria [src] 
     2                                
- cos (x) + x*cos(2*x) - cos(x)*sin(x)
--------------------------------------
           cos(x) + sin(x)
$$\frac{x \cos{\left(2 x \right)} - \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}$$ 
(-cos(x)^2 + x*cos(2*x) - cos(x)*sin(x))/(cos(x) + sin(x))
Denominador racional [src] 
x*cos(2*x) - (cos(x) + sin(x))*cos(x)
-------------------------------------
           cos(x) + sin(x)
$$\frac{x \cos{\left(2 x \right)} - \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}$$ 
(x*cos(2*x) - (cos(x) + sin(x))*cos(x))/(cos(x) + sin(x))
Abrimos la expresión [src] 
                                     2     
                 x            2*x*cos (x)  
-cos(x) - --------------- + ---------------
          cos(x) + sin(x)   cos(x) + sin(x)
$$\frac{2 x \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} - \frac{x}{\sin{\left(x \right)} + \cos{\left(x \right)}} - \cos{\left(x \right)}$$ 
-cos(x) - x/(cos(x) + sin(x)) + 2*x*cos(x)^2/(cos(x) + sin(x))
Parte trigonométrica [src] 
                      ___   
       1          x*\/ 2    
- ----------- + ------------
     /pi    \      /     pi\
  csc|-- - x|   csc|-x + --|
     \2     /      \     4 /
$$\frac{\sqrt{2} x}{\csc{\left(- x + \frac{\pi}{4} \right)}} - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}$$ 
              ___    2/x   pi\ /        2/x   pi\\
-cos(x) + x*\/ 2 *sin |- + --|*|-1 + cot |- + --||
                      \2   8 / \         \2   8 //
$$\sqrt{2} x \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - \cos{\left(x \right)}$$ 
          2/x\                                             
  -1 + cot |-|                  /        2   \             
           \2/                x*\-1 + cot (x)/             
- ------------ + ------------------------------------------
         2/x\                  /        2/x\          /x\ \
  1 + cot |-|                  |-1 + cot |-|     2*cot|-| |
          \2/    /       2   \ |         \2/          \2/ |
                 \1 + cot (x)/*|------------ + -----------|
                               |       2/x\           2/x\|
                               |1 + cot |-|    1 + cot |-||
                               \        \2/            \2//
$$\frac{x \left(\cot^{2}{\left(x \right)} - 1\right)}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(x \right)} + 1\right)} - \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}$$ 
                    ___    /    pi\
                x*\/ 2 *csc|x + --|
       1                   \    4 /
- ----------- + -------------------
     /pi    \          /pi      \  
  csc|-- - x|     2*csc|-- - 2*x|  
     \2     /          \2       /
$$\frac{\sqrt{2} x \csc{\left(x + \frac{\pi}{4} \right)}}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}$$ 
    1                  x             
- ------ + --------------------------
  sec(x)   /  1        1   \         
           |------ + ------|*sec(2*x)
           \csc(x)   sec(x)/
$$\frac{x}{\left(\frac{1}{\sec{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(2 x \right)}} - \frac{1}{\sec{\left(x \right)}}$$ 
              ___    2    /       2/x   pi\\ /       2   \
          x*\/ 2 *cos (x)*|1 + tan |- + --||*\1 - tan (x)/
                          \        \2   8 //              
-cos(x) + ------------------------------------------------
                                /x   pi\                  
                           4*tan|- + --|                  
                                \2   8 /
$$\frac{\sqrt{2} x \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(x \right)}}{4 \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}} - \cos{\left(x \right)}$$ 
         2/x\                                            
  1 - tan |-|                  /       2   \             
          \2/                x*\1 - tan (x)/             
- ----------- + -----------------------------------------
         2/x\                 /       2/x\          /x\ \
  1 + tan |-|                 |1 - tan |-|     2*tan|-| |
          \2/   /       2   \ |        \2/          \2/ |
                \1 + tan (x)/*|----------- + -----------|
                              |       2/x\          2/x\|
                              |1 + tan |-|   1 + tan |-||
                              \        \2/           \2//
$$\frac{x \left(1 - \tan^{2}{\left(x \right)}\right)}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)} - \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$ 
          2/x\       ___ /        2/x   pi\\
  -1 + cot |-|   x*\/ 2 *|-1 + cot |- + --||
           \2/           \         \2   8 //
- ------------ + ---------------------------
         2/x\                 2/x   pi\     
  1 + cot |-|          1 + cot |- + --|     
          \2/                  \2   8 /
$$\frac{\sqrt{2} x \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} - \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}$$ 
          2/x\       ___ /       2/x   pi\\ /        2   \
  -1 + cot |-|   x*\/ 2 *|1 + cot |- + --||*\-1 + cot (x)/
           \2/           \        \2   8 //               
- ------------ + -----------------------------------------
         2/x\             /       2   \    /x   pi\       
  1 + cot |-|           4*\1 + cot (x)/*cot|- + --|       
          \2/                              \2   8 /
$$\frac{\sqrt{2} x \left(\cot^{2}{\left(x \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}} - \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}$$ 
               x*cos(2*x)     
-cos(x) + --------------------
                      /    pi\
          cos(x) + cos|x - --|
                      \    2 /
$$\frac{x \cos{\left(2 x \right)}}{\cos{\left(x \right)} + \cos{\left(x - \frac{\pi}{2} \right)}} - \cos{\left(x \right)}$$ 
                       /pi      \   
                  x*sin|-- + 2*x|   
     /    pi\          \2       /   
- sin|x + --| + --------------------
     \    2 /               /    pi\
                sin(x) + sin|x + --|
                            \    2 /
$$\frac{x \sin{\left(2 x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)} + \sin{\left(x + \frac{\pi}{2} \right)}} - \sin{\left(x + \frac{\pi}{2} \right)}$$ 
         2/x\       ___ /       2/x   pi\\
  1 - tan |-|   x*\/ 2 *|1 - tan |- + --||
          \2/           \        \2   8 //
- ----------- + --------------------------
         2/x\               2/x   pi\     
  1 + tan |-|        1 + tan |- + --|     
          \2/                \2   8 /
$$\frac{\sqrt{2} x \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} - \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$ 
                    ___    /pi      \
                x*\/ 2 *sin|-- + 2*x|
     /    pi\              \2       /
- sin|x + --| + ---------------------
     \    2 /            /    pi\    
                    2*sin|x + --|    
                         \    4 /
$$\frac{\sqrt{2} x \sin{\left(2 x + \frac{\pi}{2} \right)}}{2 \sin{\left(x + \frac{\pi}{4} \right)}} - \sin{\left(x + \frac{\pi}{2} \right)}$$ 
              ___         
          x*\/ 2 *cos(2*x)
-cos(x) + ----------------
                /    pi\  
           2*sin|x + --|  
                \    4 /
$$\frac{\sqrt{2} x \cos{\left(2 x \right)}}{2 \sin{\left(x + \frac{\pi}{4} \right)}} - \cos{\left(x \right)}$$ 
              ___    /    pi\
-cos(x) + x*\/ 2 *cos|x + --|
                     \    4 /
$$\sqrt{2} x \cos{\left(x + \frac{\pi}{4} \right)} - \cos{\left(x \right)}$$ 
       1                         x                  
- ----------- + ------------------------------------
     /pi    \   /  1           1     \    /pi      \
  csc|-- - x|   |------ + -----------|*csc|-- - 2*x|
     \2     /   |csc(x)      /pi    \|    \2       /
                |         csc|-- - x||              
                \            \2     //
$$\frac{x}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}$$ 
              ___             /    pi\
          x*\/ 2 *cos(2*x)*csc|x + --|
                              \    4 /
-cos(x) + ----------------------------
                       2
$$\frac{\sqrt{2} x \cos{\left(2 x \right)} \csc{\left(x + \frac{\pi}{4} \right)}}{2} - \cos{\left(x \right)}$$ 
              ___             /    pi\
          x*\/ 2 *cos(2*x)*sec|x - --|
                              \    4 /
-cos(x) + ----------------------------
                       2
$$\frac{\sqrt{2} x \cos{\left(2 x \right)} \sec{\left(x - \frac{\pi}{4} \right)}}{2} - \cos{\left(x \right)}$$ 
               ___    /    pi\
           x*\/ 2 *sec|x - --|
    1                 \    4 /
- ------ + -------------------
  sec(x)        2*sec(2*x)
$$\frac{\sqrt{2} x \sec{\left(x - \frac{\pi}{4} \right)}}{2 \sec{\left(2 x \right)}} - \frac{1}{\sec{\left(x \right)}}$$ 
              ___ /       2/x   pi\\
          x*\/ 2 *|1 - tan |- + --||
                  \        \2   8 //
-cos(x) + --------------------------
                      2/x   pi\     
               1 + tan |- + --|     
                       \2   8 /
$$\frac{\sqrt{2} x \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} - \cos{\left(x \right)}$$ 
    1                     x               
- ------ + -------------------------------
  sec(x)   /  1           1     \         
           |------ + -----------|*sec(2*x)
           |sec(x)      /    pi\|         
           |         sec|x - --||         
           \            \    2 //
$$\frac{x}{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(2 x \right)}} - \frac{1}{\sec{\left(x \right)}}$$ 
         2/x\       ___ /       2/x   pi\\ /       2   \
  1 - tan |-|   x*\/ 2 *|1 + tan |- + --||*\1 - tan (x)/
          \2/           \        \2   8 //              
- ----------- + ----------------------------------------
         2/x\           /       2   \    /x   pi\       
  1 + tan |-|         4*\1 + tan (x)/*tan|- + --|       
          \2/                            \2   8 /
$$\frac{\sqrt{2} x \left(1 - \tan^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}} - \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$ 
     /    pi\       ___    /    3*pi\
- sin|x + --| + x*\/ 2 *sin|x + ----|
     \    2 /              \     4  /
$$\sqrt{2} x \sin{\left(x + \frac{3 \pi}{4} \right)} - \sin{\left(x + \frac{\pi}{2} \right)}$$ 
              ___    2    /        2   \    /x   pi\
          x*\/ 2 *sin (x)*\-1 + cot (x)/*tan|- + --|
                                            \2   8 /
-cos(x) + ------------------------------------------
                             2/x   pi\              
                        4*sin |- + --|              
                              \2   8 /
$$\frac{\sqrt{2} x \left(\cot^{2}{\left(x \right)} - 1\right) \sin^{2}{\left(x \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{4 \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}} - \cos{\left(x \right)}$$ 
                 ___  
    1        x*\/ 2   
- ------ + -----------
  sec(x)      /    pi\
           sec|x + --|
              \    4 /
$$\frac{\sqrt{2} x}{\sec{\left(x + \frac{\pi}{4} \right)}} - \frac{1}{\sec{\left(x \right)}}$$ 
              ___         
          x*\/ 2 *cos(2*x)
-cos(x) + ----------------
                /    pi\  
           2*cos|x - --|  
                \    4 /
$$\frac{\sqrt{2} x \cos{\left(2 x \right)}}{2 \cos{\left(x - \frac{\pi}{4} \right)}} - \cos{\left(x \right)}$$ 
-cos(x) + x*sqrt(2)*cos(2*x)/(2*cos(x - pi/4))
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