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

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

Expresión a simplificar:

Solución

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