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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(x-sin(x))/(4*x-pi)

¿Cómo vas a descomponer esta cos(x-sin(x))/(4*x-pi) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
cos(x - sin(x))
---------------
    4*x - pi
cos⁡(x−sin⁡(x))4x−π\frac{\cos{\left(x - \sin{\left(x \right)} \right)}}{4 x - \pi} 
cos(x - sin(x))/(4*x - pi)
Respuesta numérica [src] 
cos(x - sin(x))/(-3.14159265358979 + 4.0*x)
cos(x - sin(x))/(-3.14159265358979 + 4.0*x)
Potencias [src] 
   /      /   -I*x    I*x\\      /       /   -I*x    I*x\\
   |    I*\- e     + e   /|      |     I*\- e     + e   /|
 I*|x + ------------------|    I*|-x - ------------------|
   \            2         /      \             2         /
e                             e                           
--------------------------- + ----------------------------
             2                             2              
----------------------------------------------------------
                        -pi + 4*x
ei(−x−i(eix−e−ix)2)2+ei(x+i(eix−e−ix)2)24x−π\frac{\frac{e^{i \left(- x - \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right)}}{2} + \frac{e^{i \left(x + \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right)}}{2}}{4 x - \pi} 
(exp(i*(x + i*(-exp(-i*x) + exp(i*x))/2))/2 + exp(i*(-x - i*(-exp(-i*x) + exp(i*x))/2))/2)/(-pi + 4*x)
Abrimos la expresión [src] 
cos(x)*cos(sin(x))   sin(x)*sin(sin(x))
------------------ + ------------------
     4*x - pi             4*x - pi
sin⁡(x)sin⁡(sin⁡(x))4x−π+cos⁡(x)cos⁡(sin⁡(x))4x−π\frac{\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)}}{4 x - \pi} + \frac{\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{4 x - \pi} 
cos(x)*cos(sin(x))/(4*x - pi) + sin(x)*sin(sin(x))/(4*x - pi)
Parte trigonométrica [src] 
               1                
--------------------------------
               /         1     \
(-pi + 4*x)*sec|x - -----------|
               |       /pi    \|
               |    sec|-- - x||
               \       \2     //
1(4x−π)sec⁡(x−1sec⁡(−x+π2))\frac{1}{\left(4 x - \pi\right) \sec{\left(x - \frac{1}{\sec{\left(- x + \frac{\pi}{2} \right)}} \right)}} 
   /       /    pi\\
cos|x - cos|x - --||
   \       \    2 //
--------------------
     -pi + 4*x
cos⁡(x−cos⁡(x−π2))4x−π\frac{\cos{\left(x - \cos{\left(x - \frac{\pi}{2} \right)} \right)}}{4 x - \pi} 
   /    pi         \
sin|x + -- - sin(x)|
   \    2          /
--------------------
     -pi + 4*x
sin⁡(x−sin⁡(x)+π2)4x−π\frac{\sin{\left(x - \sin{\left(x \right)} + \frac{\pi}{2} \right)}}{4 x - \pi} 
                /          /x\  \      
                |       cot|-|  |      
               2|x         \2/  |      
       -1 + cot |- - -----------|      
                |2          2/x\|      
                |    1 + cot |-||      
                \            \2//      
---------------------------------------
/        /          /x\  \\            
|        |       cot|-|  ||            
|       2|x         \2/  ||            
|1 + cot |- - -----------||*(-pi + 4*x)
|        |2          2/x\||            
|        |    1 + cot |-|||            
\        \            \2///
cot⁡2(x2−cot⁡(x2)cot⁡2(x2)+1)−1(4x−π)(cot⁡2(x2−cot⁡(x2)cot⁡2(x2)+1)+1)\frac{\cot^{2}{\left(\frac{x}{2} - \frac{\cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)} - 1}{\left(4 x - \pi\right) \left(\cot^{2}{\left(\frac{x}{2} - \frac{\cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)} + 1\right)} 
               /          /x\  \       
               |       tan|-|  |       
              2|x         \2/  |       
       1 - tan |- - -----------|       
               |2          2/x\|       
               |    1 + tan |-||       
               \            \2//       
---------------------------------------
/        /          /x\  \\            
|        |       tan|-|  ||            
|       2|x         \2/  ||            
|1 + tan |- - -----------||*(-pi + 4*x)
|        |2          2/x\||            
|        |    1 + tan |-|||            
\        \            \2///
1−tan⁡2(x2−tan⁡(x2)tan⁡2(x2)+1)(4x−π)(tan⁡2(x2−tan⁡(x2)tan⁡2(x2)+1)+1)\frac{1 - \tan^{2}{\left(\frac{x}{2} - \frac{\tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{\left(4 x - \pi\right) \left(\tan^{2}{\left(\frac{x}{2} - \frac{\tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)} + 1\right)} 
               1                
--------------------------------
               /         1     \
(-pi + 4*x)*sec|x - -----------|
               |       /    pi\|
               |    sec|x - --||
               \       \    2 //
1(4x−π)sec⁡(x−1sec⁡(x−π2))\frac{1}{\left(4 x - \pi\right) \sec{\left(x - \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}} 
               1                
--------------------------------
               /  1      pi    \
(-pi + 4*x)*csc|------ + -- - x|
               \csc(x)   2     /
1(4x−π)csc⁡(−x+π2+1csc⁡(x))\frac{1}{\left(4 x - \pi\right) \csc{\left(- x + \frac{\pi}{2} + \frac{1}{\csc{\left(x \right)}} \right)}} 
             1             
---------------------------
               /      1   \
(-pi + 4*x)*sec|x - ------|
               \    csc(x)/
1(4x−π)sec⁡(x−1csc⁡(x))\frac{1}{\left(4 x - \pi\right) \sec{\left(x - \frac{1}{\csc{\left(x \right)}} \right)}} 
1/((-pi + 4*x)*sec(x - 1/csc(x)))
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