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

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

Solución

Ha introducido [src]

        sin(x)                 / sin(x)    \
cos(x)*e         (-1 - cos(x))*\E       - 1/
-------------- + ---------------------------
  x + sin(x)                        2       
                        (x + sin(x))

\frac{\left(e^{\sin{\left(x \right)}} - 1\right) \left(- \cos{\left(x \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}} + \frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{x + \sin{\left(x \right)}}

(cos(x)*exp(sin(x)))/(x + sin(x)) + ((-1 - cos(x))*(E^sin(x) - 1))/(x + sin(x))^2

Simplificación general [src]

/     sin(x)\                                     sin(x)
\1 - e      /*(1 + cos(x)) + (x + sin(x))*cos(x)*e      
--------------------------------------------------------
                                 2                      
                     (x + sin(x))

\frac{\left(1 - e^{\sin{\left(x \right)}}\right) \left(\cos{\left(x \right)} + 1\right) + \left(x + \sin{\left(x \right)}\right) e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{\left(x + \sin{\left(x \right)}\right)^{2}}

((1 - exp(sin(x)))*(1 + cos(x)) + (x + sin(x))*cos(x)*exp(sin(x)))/(x + sin(x))^2

Respuesta numérica [src]

cos(x)*exp(sin(x))/(x + sin(x)) + (-1.0 + 2.71828182845905^sin(x))*(-1.0 - cos(x))/(x + sin(x))^2

cos(x)*exp(sin(x))/(x + sin(x)) + (-1.0 + 2.71828182845905^sin(x))*(-1.0 - cos(x))/(x + sin(x))^2

Denominador común [src]

     sin(x)           sin(x)             sin(x)           sin(x)                
1 - e       - cos(x)*e       + x*cos(x)*e       + cos(x)*e      *sin(x) + cos(x)
--------------------------------------------------------------------------------
                            2      2                                            
                           x  + sin (x) + 2*x*sin(x)

\frac{x e^{\sin{\left(x \right)}} \cos{\left(x \right)} + e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} - e^{\sin{\left(x \right)}} \cos{\left(x \right)} - e^{\sin{\left(x \right)}} + \cos{\left(x \right)} + 1}{x^{2} + 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}}

(1 - exp(sin(x)) - cos(x)*exp(sin(x)) + x*cos(x)*exp(sin(x)) + cos(x)*exp(sin(x))*sin(x) + cos(x))/(x^2 + sin(x)^2 + 2*x*sin(x))

Combinatoria [src]

     sin(x)           sin(x)             sin(x)           sin(x)                
1 - e       - cos(x)*e       + x*cos(x)*e       + cos(x)*e      *sin(x) + cos(x)
--------------------------------------------------------------------------------
                                             2                                  
                                 (x + sin(x))

\frac{x e^{\sin{\left(x \right)}} \cos{\left(x \right)} + e^{\sin{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} - e^{\sin{\left(x \right)}} \cos{\left(x \right)} - e^{\sin{\left(x \right)}} + \cos{\left(x \right)} + 1}{\left(x + \sin{\left(x \right)}\right)^{2}}

(1 - exp(sin(x)) - cos(x)*exp(sin(x)) + x*cos(x)*exp(sin(x)) + cos(x)*exp(sin(x))*sin(x) + cos(x))/(x + sin(x))^2

Potencias [src]

                   /   -I*x    I*x\    /         /   -I*x    I*x\ \                    
                -I*\- e     + e   /    |      -I*\- e     + e   / |                    
/ I*x    -I*x\  --------------------   |      --------------------| /      I*x    -I*x\
|e      e    |           2             |               2          | |     e      e    |
|---- + -----|*e                       \-1 + e                    /*|-1 - ---- - -----|
\ 2       2  /                                                      \      2       2  /
------------------------------------ + ------------------------------------------------
             /   -I*x    I*x\                                             2            
           I*\- e     + e   /                     /      /   -I*x    I*x\\             
       x - ------------------                     |    I*\- e     + e   /|             
                   2                              |x - ------------------|             
                                                  \            2         /

\frac{\left(-1 + e^{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2}}\right) \left(- \frac{e^{i x}}{2} - 1 - \frac{e^{- i x}}{2}\right)}{\left(x - \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right)^{2}} + \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) e^{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2}}}{x - \frac{i \left(e^{i x} - e^{- i x}\right)}{2}}

        sin(x)                 /      sin(x)\
cos(x)*e         (-1 - cos(x))*\-1 + e      /
-------------- + ----------------------------
  x + sin(x)                        2        
                        (x + sin(x))

\frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{\left(e^{\sin{\left(x \right)}} - 1\right) \left(- \cos{\left(x \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

cos(x)*exp(sin(x))/(x + sin(x)) + (-1 - cos(x))*(-1 + exp(sin(x)))/(x + sin(x))^2

Denominador racional [src]

            2         sin(x)                 /      sin(x)\             
(x + sin(x)) *cos(x)*e       + (-1 - cos(x))*\-1 + e      /*(x + sin(x))
------------------------------------------------------------------------
                                         3                              
                             (x + sin(x))

\frac{\left(x + \sin{\left(x \right)}\right)^{2} e^{\sin{\left(x \right)}} \cos{\left(x \right)} + \left(x + \sin{\left(x \right)}\right) \left(e^{\sin{\left(x \right)}} - 1\right) \left(- \cos{\left(x \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{3}}

((x + sin(x))^2*cos(x)*exp(sin(x)) + (-1 - cos(x))*(-1 + exp(sin(x)))*(x + sin(x)))/(x + sin(x))^3

Unión de expresiones racionales [src]

              /      sin(x)\                        sin(x)
(-1 - cos(x))*\-1 + e      / + (x + sin(x))*cos(x)*e      
----------------------------------------------------------
                                  2                       
                      (x + sin(x))

\frac{\left(x + \sin{\left(x \right)}\right) e^{\sin{\left(x \right)}} \cos{\left(x \right)} + \left(e^{\sin{\left(x \right)}} - 1\right) \left(- \cos{\left(x \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

((-1 - cos(x))*(-1 + exp(sin(x))) + (x + sin(x))*cos(x)*exp(sin(x)))/(x + sin(x))^2

Abrimos la expresión [src]

                                                                sin(x)             sin(x)                          sin(x)     
            1                         cos(x)            cos(x)*e                  e                        cos(x)*e           
------------------------- + ------------------------- + -------------- - ------------------------- - -------------------------
 2      2                    2      2                     x + sin(x)      2      2                    2      2                
x  + sin (x) + 2*x*sin(x)   x  + sin (x) + 2*x*sin(x)                    x  + sin (x) + 2*x*sin(x)   x  + sin (x) + 2*x*sin(x)

\frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{x + \sin{\left(x \right)}} - \frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{x^{2} + 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}} - \frac{e^{\sin{\left(x \right)}}}{x^{2} + 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{x^{2} + 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}} + \frac{1}{x^{2} + 2 x \sin{\left(x \right)} + \sin^{2}{\left(x \right)}}

1/(x^2 + sin(x)^2 + 2*x*sin(x)) + cos(x)/(x^2 + sin(x)^2 + 2*x*sin(x)) + (cos(x)*exp(sin(x)))/(x + sin(x)) - exp(sin(x))/(x^2 + sin(x)^2 + 2*x*sin(x)) - cos(x)*exp(sin(x))/(x^2 + sin(x)^2 + 2*x*sin(x))

Parte trigonométrica [src]

                                                    /                        sin(x)\
(cosh(sin(x)) + sinh(sin(x)))*cos(x)   (1 + cos(x))*\-1 + (cosh(1) + sinh(1))      /
------------------------------------ - ---------------------------------------------
             x + sin(x)                                            2                
                                                       (x + sin(x))

\frac{\left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)}}{x + \sin{\left(x \right)}} - \frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\sin{\left(x \right)}} - 1\right) \left(\cos{\left(x \right)} + 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

                    /                               /x\ \                                                        
                    |                          2*tan|-| |                                                        
                    |                               \2/ |                                                        
                    |                        -----------|                                                        
  /           2/x\\ |                               2/x\|                 /    /       /x\ \       /       /x\ \\
  |    1 - tan |-|| |                        1 + tan |-||                 |    |  2*tan|-| |       |  2*tan|-| ||
  |            \2/| |                                \2/|   /       2/x\\ |    |       \2/ |       |       \2/ ||
  |1 + -----------|*\-1 + (cosh(1) + sinh(1))           /   |1 - tan |-||*|cosh|-----------| + sinh|-----------||
  |           2/x\|                                         \        \2// |    |       2/x\|       |       2/x\||
  |    1 + tan |-||                                                       |    |1 + tan |-||       |1 + tan |-|||
  \            \2//                                                       \    \        \2//       \        \2///
- ------------------------------------------------------- + -----------------------------------------------------
                                      2                                              /           /x\ \           
                     /           /x\ \                                               |      2*tan|-| |           
                     |      2*tan|-| |                                 /       2/x\\ |           \2/ |           
                     |           \2/ |                                 |1 + tan |-||*|x + -----------|           
                     |x + -----------|                                 \        \2// |           2/x\|           
                     |           2/x\|                                               |    1 + tan |-||           
                     |    1 + tan |-||                                               \            \2//           
                     \            \2//

\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\sinh{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)} + \cosh{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}\right)}{\left(x + \frac{2 \tan{\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{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} - 1\right)}{\left(x + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}

                                       /                        sin(x)\              
(cosh(sin(x)) + sinh(sin(x)))*cos(x)   \-1 + (cosh(1) + sinh(1))      /*(-1 - cos(x))
------------------------------------ + ----------------------------------------------
             x + sin(x)                                            2                 
                                                       (x + sin(x))

\frac{\left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\sin{\left(x \right)}} - 1\right) \left(- \cos{\left(x \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

(cosh(sin(x)) + sinh(sin(x)))*cos(x)   (1 + cos(x))*(-1 + cosh(sin(x)) + sinh(sin(x)))
------------------------------------ - -----------------------------------------------
             x + sin(x)                                             2                 
                                                        (x + sin(x))

\frac{\left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)}}{x + \sin{\left(x \right)}} - \frac{\left(\cos{\left(x \right)} + 1\right) \left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

                                 /    pi\   /       /    pi\\ /                        sin(x)\
(cosh(sin(x)) + sinh(sin(x)))*sin|x + --|   |1 + sin|x + --||*\-1 + (cosh(1) + sinh(1))      /
                                 \    2 /   \       \    2 //                                 
----------------------------------------- - --------------------------------------------------
                x + sin(x)                                                2                   
                                                              (x + sin(x))

\frac{\left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)}\right) \sin{\left(x + \frac{\pi}{2} \right)}}{x + \sin{\left(x \right)}} - \frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\sin{\left(x \right)}} - 1\right) \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

                                                 /                           /    pi\\              
/    /   /    pi\\       /   /    pi\\\          |                        cos|x - --||              
|cosh|cos|x - --|| + sinh|cos|x - --|||*cos(x)   |                           \    2 /|              
\    \   \    2 //       \   \    2 ///          \-1 + (cosh(1) + sinh(1))           /*(-1 - cos(x))
---------------------------------------------- + ---------------------------------------------------
                      /    pi\                                                     2                
               x + cos|x - --|                                    /       /    pi\\                 
                      \    2 /                                    |x + cos|x - --||                 
                                                                  \       \    2 //

\frac{\left(\sinh{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)} + \cosh{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)}\right) \cos{\left(x \right)}}{x + \cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\cos{\left(x - \frac{\pi}{2} \right)}} - 1\right) \left(- \cos{\left(x \right)} - 1\right)}{\left(x + \cos{\left(x - \frac{\pi}{2} \right)}\right)^{2}}

                                                     2/x\ /        2/x\\                              
               /                        sin(x)\   cos |-|*|-1 + tan |-||*(cosh(sin(x)) + sinh(sin(x)))
  (1 + cos(x))*\-1 + (cosh(1) + sinh(1))      /       \2/ \         \2//                              
- --------------------------------------------- - ----------------------------------------------------
                              2                                        x + sin(x)                     
                  (x + sin(x))

- \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)}\right) \cos^{2}{\left(\frac{x}{2} \right)}}{x + \sin{\left(x \right)}} - \frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\sin{\left(x \right)}} - 1\right) \left(\cos{\left(x \right)} + 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

                   /             /x\ \                                  
                   |        2*tan|-| |                                  
                   |             \2/ |                            /x\   
                   |      -----------|                       2*tan|-|   
/            2/x\\ |             2/x\|                            \2/   
|     1 - tan |-|| |      1 + tan |-||                     -----------  
|             \2/| |              \2/|                            2/x\  
|-1 - -----------|*\-1 + e           /                     1 + tan |-|  
|            2/x\|                          /       2/x\\          \2/  
|     1 + tan |-||                          |1 - tan |-||*e             
\             \2//                          \        \2//               
-------------------------------------- + -------------------------------
                           2                           /           /x\ \
          /           /x\ \                            |      2*tan|-| |
          |      2*tan|-| |              /       2/x\\ |           \2/ |
          |           \2/ |              |1 + tan |-||*|x + -----------|
          |x + -----------|              \        \2// |           2/x\|
          |           2/x\|                            |    1 + tan |-||
          |    1 + tan |-||                            \            \2//
          \            \2//

\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) e^{\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}}{\left(x + \frac{2 \tan{\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{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right) \left(e^{\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} - 1\right)}{\left(x + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}

        sin(x)                /      sin(x)\
cos(x)*e         (1 + cos(x))*\-1 + e      /
-------------- - ---------------------------
  x + sin(x)                        2       
                        (x + sin(x))

\frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{x + \sin{\left(x \right)}} - \frac{\left(e^{\sin{\left(x \right)}} - 1\right) \left(\cos{\left(x \right)} + 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

                                              /        1   \
                                              |      ------|
                           /          1     \ |      csc(x)|
           1               |-1 - -----------|*\-1 + e      /
         ------            |        /pi    \|               
         csc(x)            |     csc|-- - x||               
        e                  \        \2     //               
------------------------ + ---------------------------------
/      1   \    /pi    \                         2          
|x + ------|*csc|-- - x|             /      1   \           
\    csc(x)/    \2     /             |x + ------|           
                                     \    csc(x)/

\frac{\left(-1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(e^{\frac{1}{\csc{\left(x \right)}}} - 1\right)}{\left(x + \frac{1}{\csc{\left(x \right)}}\right)^{2}} + \frac{e^{\frac{1}{\csc{\left(x \right)}}}}{\left(x + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}

                                                              /                           /    pi\\
/    /   /    pi\\       /   /    pi\\\                       |                        cos|x - --||
|cosh|cos|x - --|| + sinh|cos|x - --|||*cos(x)                |                           \    2 /|
\    \   \    2 //       \   \    2 ///          (1 + cos(x))*\-1 + (cosh(1) + sinh(1))           /
---------------------------------------------- - --------------------------------------------------
                      /    pi\                                                    2                
               x + cos|x - --|                                   /       /    pi\\                 
                      \    2 /                                   |x + cos|x - --||                 
                                                                 \       \    2 //

\frac{\left(\sinh{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)} + \cosh{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)}\right) \cos{\left(x \right)}}{x + \cos{\left(x - \frac{\pi}{2} \right)}} - \frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\cos{\left(x - \frac{\pi}{2} \right)}} - 1\right) \left(\cos{\left(x \right)} + 1\right)}{\left(x + \cos{\left(x - \frac{\pi}{2} \right)}\right)^{2}}

                                                     2/x\ /        2/x\\                              
               /                        sin(x)\   sin |-|*|-1 + cot |-||*(cosh(sin(x)) + sinh(sin(x)))
  (1 + cos(x))*\-1 + (cosh(1) + sinh(1))      /       \2/ \         \2//                              
- --------------------------------------------- + ----------------------------------------------------
                              2                                        x + sin(x)                     
                  (x + sin(x))

\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)}\right) \sin^{2}{\left(\frac{x}{2} \right)}}{x + \sin{\left(x \right)}} - \frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\sin{\left(x \right)}} - 1\right) \left(\cos{\left(x \right)} + 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

/    /   /    pi\\       /   /    pi\\\                       /         /   /    pi\\       /   /    pi\\\
|cosh|cos|x - --|| + sinh|cos|x - --|||*cos(x)   (1 + cos(x))*|-1 + cosh|cos|x - --|| + sinh|cos|x - --|||
\    \   \    2 //       \   \    2 ///                       \         \   \    2 //       \   \    2 ///
---------------------------------------------- - ---------------------------------------------------------
                      /    pi\                                                        2                   
               x + cos|x - --|                                       /       /    pi\\                    
                      \    2 /                                       |x + cos|x - --||                    
                                                                     \       \    2 //

\frac{\left(\sinh{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)} + \cosh{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)}\right) \cos{\left(x \right)}}{x + \cos{\left(x - \frac{\pi}{2} \right)}} - \frac{\left(\cos{\left(x \right)} + 1\right) \left(\sinh{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)} + \cosh{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)} - 1\right)}{\left(x + \cos{\left(x - \frac{\pi}{2} \right)}\right)^{2}}

                                    /        1   \
         1                          |      ------|
       ------         /       1   \ |      csc(x)|
       csc(x)         |-1 - ------|*\-1 + e      /
      e               \     sec(x)/               
------------------- + ----------------------------
/      1   \                             2        
|x + ------|*sec(x)          /      1   \         
\    csc(x)/                 |x + ------|         
                             \    csc(x)/

\frac{\left(-1 - \frac{1}{\sec{\left(x \right)}}\right) \left(e^{\frac{1}{\csc{\left(x \right)}}} - 1\right)}{\left(x + \frac{1}{\csc{\left(x \right)}}\right)^{2}} + \frac{e^{\frac{1}{\csc{\left(x \right)}}}}{\left(x + \frac{1}{\csc{\left(x \right)}}\right) \sec{\left(x \right)}}

                                 /    pi\   /                        sin(x)\ /        /    pi\\
(cosh(sin(x)) + sinh(sin(x)))*sin|x + --|   \-1 + (cosh(1) + sinh(1))      /*|-1 - sin|x + --||
                                 \    2 /                                    \        \    2 //
----------------------------------------- + ---------------------------------------------------
                x + sin(x)                                                 2                   
                                                               (x + sin(x))

\frac{\left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)}\right) \sin{\left(x + \frac{\pi}{2} \right)}}{x + \sin{\left(x \right)}} + \frac{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\sin{\left(x \right)}} - 1\right) \left(- \sin{\left(x + \frac{\pi}{2} \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

                                         /           1     \
            1                            |      -----------|
       -----------                       |         /    pi\|
          /    pi\                       |      sec|x - --||
       sec|x - --|         /       1   \ |         \    2 /|
          \    2 /         |-1 - ------|*\-1 + e           /
      e                    \     sec(x)/                    
------------------------ + ---------------------------------
/         1     \                                   2       
|x + -----------|*sec(x)           /         1     \        
|       /    pi\|                  |x + -----------|        
|    sec|x - --||                  |       /    pi\|        
\       \    2 //                  |    sec|x - --||        
                                   \       \    2 //

\frac{\left(-1 - \frac{1}{\sec{\left(x \right)}}\right) \left(e^{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} - 1\right)}{\left(x + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}} + \frac{e^{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}}}{\left(x + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x \right)}}

                                 /    pi\   /       /    pi\\                                   
(cosh(sin(x)) + sinh(sin(x)))*sin|x + --|   |1 + sin|x + --||*(-1 + cosh(sin(x)) + sinh(sin(x)))
                                 \    2 /   \       \    2 //                                   
----------------------------------------- - ----------------------------------------------------
                x + sin(x)                                                 2                    
                                                               (x + sin(x))

\frac{\left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)}\right) \sin{\left(x + \frac{\pi}{2} \right)}}{x + \sin{\left(x \right)}} - \frac{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right) \left(\sinh{\left(\sin{\left(x \right)} \right)} + \cosh{\left(\sin{\left(x \right)} \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

                    /             /x\ \                                  
                    |        2*tan|-| |                                  
                    |             \2/ |                            /x\   
                    |      -----------|                       2*tan|-|   
  /           2/x\\ |             2/x\|                            \2/   
  |    1 - tan |-|| |      1 + tan |-||                     -----------  
  |            \2/| |              \2/|                            2/x\  
  |1 + -----------|*\-1 + e           /                     1 + tan |-|  
  |           2/x\|                          /       2/x\\          \2/  
  |    1 + tan |-||                          |1 - tan |-||*e             
  \            \2//                          \        \2//               
- ------------------------------------- + -------------------------------
                             2                          /           /x\ \
            /           /x\ \                           |      2*tan|-| |
            |      2*tan|-| |             /       2/x\\ |           \2/ |
            |           \2/ |             |1 + tan |-||*|x + -----------|
            |x + -----------|             \        \2// |           2/x\|
            |           2/x\|                           |    1 + tan |-||
            |    1 + tan |-||                           \            \2//
            \            \2//

\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) e^{\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}}{\left(x + \frac{2 \tan{\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{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(e^{\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} - 1\right)}{\left(x + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}

                                        /           1     \
            1                           |      -----------|
       -----------                      |         /    pi\|
          /    pi\                      |      sec|x - --||
       sec|x - --|         /      1   \ |         \    2 /|
          \    2 /         |1 + ------|*\-1 + e           /
      e                    \    sec(x)/                    
------------------------ - --------------------------------
/         1     \                                  2       
|x + -----------|*sec(x)          /         1     \        
|       /    pi\|                 |x + -----------|        
|    sec|x - --||                 |       /    pi\|        
\       \    2 //                 |    sec|x - --||        
                                  \       \    2 //

- \frac{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \left(e^{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} - 1\right)}{\left(x + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}} + \frac{e^{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}}}{\left(x + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x \right)}}

                    /             /x\ \                                  
                    |        2*cot|-| |                                  
                    |             \2/ |                            /x\   
                    |      -----------|                       2*cot|-|   
/             2/x\\ |             2/x\|                            \2/   
|     -1 + cot |-|| |      1 + cot |-||                     -----------  
|              \2/| |              \2/|                            2/x\  
|-1 - ------------|*\-1 + e           /                     1 + cot |-|  
|            2/x\ |                         /        2/x\\          \2/  
|     1 + cot |-| |                         |-1 + cot |-||*e             
\             \2/ /                         \         \2//               
--------------------------------------- + -------------------------------
                            2                           /           /x\ \
           /           /x\ \                            |      2*cot|-| |
           |      2*cot|-| |              /       2/x\\ |           \2/ |
           |           \2/ |              |1 + cot |-||*|x + -----------|
           |x + -----------|              \        \2// |           2/x\|
           |           2/x\|                            |    1 + cot |-||
           |    1 + cot |-||                            \            \2//
           \            \2//

\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) e^{\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}}{\left(x + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right) \left(e^{\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} - 1\right)}{\left(x + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}

                                             /        1   \
                                             |      ------|
                           /         1     \ |      csc(x)|
           1               |1 + -----------|*\-1 + e      /
         ------            |       /pi    \|               
         csc(x)            |    csc|-- - x||               
        e                  \       \2     //               
------------------------ - --------------------------------
/      1   \    /pi    \                        2          
|x + ------|*csc|-- - x|            /      1   \           
\    csc(x)/    \2     /            |x + ------|           
                                    \    csc(x)/

- \frac{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(e^{\frac{1}{\csc{\left(x \right)}}} - 1\right)}{\left(x + \frac{1}{\csc{\left(x \right)}}\right)^{2}} + \frac{e^{\frac{1}{\csc{\left(x \right)}}}}{\left(x + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}

                     /                               /x\ \                                                         
                     |                          2*cot|-| |                                                         
                     |                               \2/ |                                                         
                     |                        -----------|                                                         
  /            2/x\\ |                               2/x\|                  /    /       /x\ \       /       /x\ \\
  |    -1 + cot |-|| |                        1 + cot |-||                  |    |  2*cot|-| |       |  2*cot|-| ||
  |             \2/| |                                \2/|   /        2/x\\ |    |       \2/ |       |       \2/ ||
  |1 + ------------|*\-1 + (cosh(1) + sinh(1))           /   |-1 + cot |-||*|cosh|-----------| + sinh|-----------||
  |           2/x\ |                                         \         \2// |    |       2/x\|       |       2/x\||
  |    1 + cot |-| |                                                        |    |1 + cot |-||       |1 + cot |-|||
  \            \2/ /                                                        \    \        \2//       \        \2///
- -------------------------------------------------------- + ------------------------------------------------------
                                      2                                               /           /x\ \            
                     /           /x\ \                                                |      2*cot|-| |            
                     |      2*cot|-| |                                  /       2/x\\ |           \2/ |            
                     |           \2/ |                                  |1 + cot |-||*|x + -----------|            
                     |x + -----------|                                  \        \2// |           2/x\|            
                     |           2/x\|                                                |    1 + cot |-||            
                     |    1 + cot |-||                                                \            \2//            
                     \            \2//

\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\sinh{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)} + \cosh{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}\right)}{\left(x + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} - 1\right)}{\left(x + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}

                                                /                          1   \
                                                |                        ------|
                              /         1     \ |                        csc(x)|
                              |1 + -----------|*\-1 + (cosh(1) + sinh(1))      /
    /  1   \       /  1   \   |       /pi    \|                                 
cosh|------| + sinh|------|   |    csc|-- - x||                                 
    \csc(x)/       \csc(x)/   \       \2     //                                 
--------------------------- - --------------------------------------------------
  /      1   \    /pi    \                                  2                   
  |x + ------|*csc|-- - x|                      /      1   \                    
  \    csc(x)/    \2     /                      |x + ------|                    
                                                \    csc(x)/

- \frac{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\frac{1}{\csc{\left(x \right)}}} - 1\right)}{\left(x + \frac{1}{\csc{\left(x \right)}}\right)^{2}} + \frac{\sinh{\left(\frac{1}{\csc{\left(x \right)}} \right)} + \cosh{\left(\frac{1}{\csc{\left(x \right)}} \right)}}{\left(x + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}

                                                     /                             1     \
                                                     |                        -----------|
    /     1     \       /     1     \                |                           /    pi\|
cosh|-----------| + sinh|-----------|                |                        sec|x - --||
    |   /    pi\|       |   /    pi\|   /      1   \ |                           \    2 /|
    |sec|x - --||       |sec|x - --||   |1 + ------|*\-1 + (cosh(1) + sinh(1))           /
    \   \    2 //       \   \    2 //   \    sec(x)/                                      
------------------------------------- - --------------------------------------------------
       /         1     \                                                 2                
       |x + -----------|*sec(x)                         /         1     \                 
       |       /    pi\|                                |x + -----------|                 
       |    sec|x - --||                                |       /    pi\|                 
       \       \    2 //                                |    sec|x - --||                 
                                                        \       \    2 //

- \frac{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} - 1\right)}{\left(x + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}} + \frac{\sinh{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)} + \cosh{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{\left(x + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x \right)}}

                     /             /x\ \                                  
                     |        2*cot|-| |                                  
                     |             \2/ |                            /x\   
                     |      -----------|                       2*cot|-|   
  /            2/x\\ |             2/x\|                            \2/   
  |    -1 + cot |-|| |      1 + cot |-||                     -----------  
  |             \2/| |              \2/|                            2/x\  
  |1 + ------------|*\-1 + e           /                     1 + cot |-|  
  |           2/x\ |                         /        2/x\\          \2/  
  |    1 + cot |-| |                         |-1 + cot |-||*e             
  \            \2/ /                         \         \2//               
- -------------------------------------- + -------------------------------
                             2                           /           /x\ \
            /           /x\ \                            |      2*cot|-| |
            |      2*cot|-| |              /       2/x\\ |           \2/ |
            |           \2/ |              |1 + cot |-||*|x + -----------|
            |x + -----------|              \        \2// |           2/x\|
            |           2/x\|                            |    1 + cot |-||
            |    1 + cot |-||                            \            \2//
            \            \2//

\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) e^{\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}}{\left(x + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(e^{\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} - 1\right)}{\left(x + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}

-(1 + (-1 + cot(x/2)^2)/(1 + cot(x/2)^2))*(-1 + exp(2*cot(x/2)/(1 + cot(x/2)^2)))/(x + 2*cot(x/2)/(1 + cot(x/2)^2))^2 + (-1 + cot(x/2)^2)*exp(2*cot(x/2)/(1 + cot(x/2)^2))/((1 + cot(x/2)^2)*(x + 2*cot(x/2)/(1 + cot(x/2)^2)))

Compilar la expresión [src]

        sin(x)                 /      sin(x)\
cos(x)*e         (-1 - cos(x))*\-1 + e      /
-------------- + ----------------------------
  x + sin(x)                        2        
                        (x + sin(x))

\frac{e^{\sin{\left(x \right)}} \cos{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{\left(e^{\sin{\left(x \right)}} - 1\right) \left(- \cos{\left(x \right)} - 1\right)}{\left(x + \sin{\left(x \right)}\right)^{2}}

cos(x)*exp(sin(x))/(x + sin(x)) + (-1 - cos(x))*(-1 + exp(sin(x)))/(x + sin(x))^2

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

Solución

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

Solución

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