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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   sin(x)    /                      sin(x) \
log      (x)*|cos(x)*log(log(x)) + --------|
             \                     x*log(x)/
(log⁡(log⁡(x))cos⁡(x)+sin⁡(x)xlog⁡(x))log⁡(x)sin⁡(x)\left(\log{\left(\log{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}}\right) \log{\left(x \right)}^{\sin{\left(x \right)}} 
log(x)^sin(x)*(cos(x)*log(log(x)) + sin(x)/((x*log(x))))
Simplificación general [src] 
   -1 + sin(x)                                          
log           (x)*(x*cos(x)*log(x)*log(log(x)) + sin(x))
--------------------------------------------------------
                           x
(xlog⁡(x)log⁡(log⁡(x))cos⁡(x)+sin⁡(x))log⁡(x)sin⁡(x)−1x\frac{\left(x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)}^{\sin{\left(x \right)} - 1}}{x} 
log(x)^(-1 + sin(x))*(x*cos(x)*log(x)*log(log(x)) + sin(x))/x
Respuesta numérica [src] 
log(x)^sin(x)*(cos(x)*log(log(x)) + sin(x)/(x*log(x)))
log(x)^sin(x)*(cos(x)*log(log(x)) + sin(x)/(x*log(x)))
Potencias [src] 
           /   -I*x    I*x\                                                   
        -I*\- e     + e   /                                                   
        -------------------- // I*x    -I*x\                 /   -I*x    I*x\\
                 2           ||e      e    |               I*\- e     + e   /|
(log(x))                    *||---- + -----|*log(log(x)) - ------------------|
                             \\ 2       2  /                   2*x*log(x)    /
((eix2+e−ix2)log⁡(log⁡(x))−i(eix−e−ix)2xlog⁡(x))log⁡(x)−i(eix−e−ix)2\left(\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \log{\left(\log{\left(x \right)} \right)} - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 x \log{\left(x \right)}}\right) \log{\left(x \right)}^{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2}} 
log(x)^(-i*(-exp(-i*x) + exp(i*x))/2)*((exp(i*x)/2 + exp(-i*x)/2)*log(log(x)) - i*(-exp(-i*x) + exp(i*x))/(2*x*log(x)))
Combinatoria [src] 
   sin(x)                                          
log      (x)*(x*cos(x)*log(x)*log(log(x)) + sin(x))
---------------------------------------------------
                      x*log(x)
(xlog⁡(x)log⁡(log⁡(x))cos⁡(x)+sin⁡(x))log⁡(x)sin⁡(x)xlog⁡(x)\frac{\left(x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)}^{\sin{\left(x \right)}}}{x \log{\left(x \right)}} 
log(x)^sin(x)*(x*cos(x)*log(x)*log(log(x)) + sin(x))/(x*log(x))
Denominador racional [src] 
   sin(x)                                          
log      (x)*(x*cos(x)*log(x)*log(log(x)) + sin(x))
---------------------------------------------------
                      x*log(x)
(xlog⁡(x)log⁡(log⁡(x))cos⁡(x)+sin⁡(x))log⁡(x)sin⁡(x)xlog⁡(x)\frac{\left(x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)}^{\sin{\left(x \right)}}}{x \log{\left(x \right)}} 
log(x)^sin(x)*(x*cos(x)*log(x)*log(log(x)) + sin(x))/(x*log(x))
Denominador común [src] 
                                     sin(x)          
   sin(x)                         log      (x)*sin(x)
log      (x)*cos(x)*log(log(x)) + -------------------
                                        x*log(x)
log⁡(x)sin⁡(x)log⁡(log⁡(x))cos⁡(x)+log⁡(x)sin⁡(x)sin⁡(x)xlog⁡(x)\log{\left(x \right)}^{\sin{\left(x \right)}} \log{\left(\log{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{\log{\left(x \right)}^{\sin{\left(x \right)}} \sin{\left(x \right)}}{x \log{\left(x \right)}} 
log(x)^sin(x)*cos(x)*log(log(x)) + log(x)^sin(x)*sin(x)/(x*log(x))
Unión de expresiones racionales [src] 
   sin(x)                                          
log      (x)*(x*cos(x)*log(x)*log(log(x)) + sin(x))
---------------------------------------------------
                      x*log(x)
(xlog⁡(x)log⁡(log⁡(x))cos⁡(x)+sin⁡(x))log⁡(x)sin⁡(x)xlog⁡(x)\frac{\left(x \log{\left(x \right)} \log{\left(\log{\left(x \right)} \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)}^{\sin{\left(x \right)}}}{x \log{\left(x \right)}} 
log(x)^sin(x)*(x*cos(x)*log(x)*log(log(x)) + sin(x))/(x*log(x))
Abrimos la expresión [src] 
                                     sin(x)          
   sin(x)                         log      (x)*sin(x)
log      (x)*cos(x)*log(log(x)) + -------------------
                                        x*log(x)
log⁡(x)sin⁡(x)log⁡(log⁡(x))cos⁡(x)+log⁡(x)sin⁡(x)sin⁡(x)xlog⁡(x)\log{\left(x \right)}^{\sin{\left(x \right)}} \log{\left(\log{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{\log{\left(x \right)}^{\sin{\left(x \right)}} \sin{\left(x \right)}}{x \log{\left(x \right)}} 
log(x)^sin(x)*cos(x)*log(log(x)) + log(x)^sin(x)*sin(x)/(x*log(x))
Parte trigonométrica [src] 
          1                                   
        ------                                
        csc(x) /log(log(x))          1       \
(log(x))      *|----------- + ---------------|
               |   /pi    \   x*csc(x)*log(x)|
               |csc|-- - x|                  |
               \   \2     /                  /
(log⁡(log⁡(x))csc⁡(−x+π2)+1xlog⁡(x)csc⁡(x))log⁡(x)1csc⁡(x)\left(\frac{\log{\left(\log{\left(x \right)} \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{x \log{\left(x \right)} \csc{\left(x \right)}}\right) \log{\left(x \right)}^{\frac{1}{\csc{\left(x \right)}}} 
             1                                          
        -----------                                     
           /    pi\                                     
        sec|x - --|                                     
           \    2 / /log(log(x))            1          \
(log(x))           *|----------- + --------------------|
                    |   sec(x)                 /    pi\|
                    |              x*log(x)*sec|x - --||
                    \                          \    2 //
(log⁡(log⁡(x))sec⁡(x)+1xlog⁡(x)sec⁡(x−π2))log⁡(x)1sec⁡(x−π2)\left(\frac{\log{\left(\log{\left(x \right)} \right)}}{\sec{\left(x \right)}} + \frac{1}{x \log{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}\right) \log{\left(x \right)}^{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} 
               /x\                                                      
          2*tan|-|                                                      
               \2/                                                      
        -----------                                                     
               2/x\ //       2/x\\                           /x\       \
        1 + tan |-| ||1 - tan |-||*log(log(x))          2*tan|-|       |
                \2/ |\        \2//                           \2/       |
(log(x))           *|------------------------- + ----------------------|
                    |              2/x\            /       2/x\\       |
                    |       1 + tan |-|          x*|1 + tan |-||*log(x)|
                    \               \2/            \        \2//       /
((1−tan⁡2(x2))log⁡(log⁡(x))tan⁡2(x2)+1+2tan⁡(x2)x(tan⁡2(x2)+1)log⁡(x))log⁡(x)2tan⁡(x2)tan⁡2(x2)+1\left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \log{\left(\log{\left(x \right)} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{x \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x \right)}}\right) \log{\left(x \right)}^{\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} 
               /x\                                                       
          2*cot|-|                                                       
               \2/                                                       
        -----------                                                      
               2/x\ //        2/x\\                           /x\       \
        1 + cot |-| ||-1 + cot |-||*log(log(x))          2*cot|-|       |
                \2/ |\         \2//                           \2/       |
(log(x))           *|-------------------------- + ----------------------|
                    |              2/x\             /       2/x\\       |
                    |       1 + cot |-|           x*|1 + cot |-||*log(x)|
                    \               \2/             \        \2//       /
((cot⁡2(x2)−1)log⁡(log⁡(x))cot⁡2(x2)+1+2cot⁡(x2)x(cot⁡2(x2)+1)log⁡(x))log⁡(x)2cot⁡(x2)cot⁡2(x2)+1\left(\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \log{\left(\log{\left(x \right)} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{x \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \log{\left(x \right)}}\right) \log{\left(x \right)}^{\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} 
           /    pi\ /                        /    pi\\
        cos|x - --| |                     cos|x - --||
           \    2 / |                        \    2 /|
(log(x))           *|cos(x)*log(log(x)) + -----------|
                    \                       x*log(x) /
(log⁡(log⁡(x))cos⁡(x)+cos⁡(x−π2)xlog⁡(x))log⁡(x)cos⁡(x−π2)\left(\log{\left(\log{\left(x \right)} \right)} \cos{\left(x \right)} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{x \log{\left(x \right)}}\right) \log{\left(x \right)}^{\cos{\left(x - \frac{\pi}{2} \right)}} 
          1                                   
        ------                                
        csc(x) /log(log(x))          1       \
(log(x))      *|----------- + ---------------|
               \   sec(x)     x*csc(x)*log(x)/
(log⁡(log⁡(x))sec⁡(x)+1xlog⁡(x)csc⁡(x))log⁡(x)1csc⁡(x)\left(\frac{\log{\left(\log{\left(x \right)} \right)}}{\sec{\left(x \right)}} + \frac{1}{x \log{\left(x \right)} \csc{\left(x \right)}}\right) \log{\left(x \right)}^{\frac{1}{\csc{\left(x \right)}}} 
   sin(x)    /               /    pi\    sin(x) \
log      (x)*|log(log(x))*sin|x + --| + --------|
             \               \    2 /   x*log(x)/
(log⁡(log⁡(x))sin⁡(x+π2)+sin⁡(x)xlog⁡(x))log⁡(x)sin⁡(x)\left(\log{\left(\log{\left(x \right)} \right)} \sin{\left(x + \frac{\pi}{2} \right)} + \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}}\right) \log{\left(x \right)}^{\sin{\left(x \right)}} 
log(x)^sin(x)*(log(log(x))*sin(x + pi/2) + sin(x)/(x*log(x)))
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