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Simplificación de expresiones/ Descomposición en fracciones simples/ cos^4x/(sinx(cos^5x+sin^5x))

¿Cómo vas a descomponer esta cos^4x/(sinx(cos^5x+sin^5x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
            4             
         cos (x)          
--------------------------
       /   5         5   \
sin(x)*\cos (x) + sin (x)/
cos⁡4(x)(sin⁡5(x)+cos⁡5(x))sin⁡(x)\frac{\cos^{4}{\left(x \right)}}{\left(\sin^{5}{\left(x \right)} + \cos^{5}{\left(x \right)}\right) \sin{\left(x \right)}} 
cos(x)^4/((sin(x)*(cos(x)^5 + sin(x)^5)))
Respuesta numérica [src] 
cos(x)^4/((cos(x)^5 + sin(x)^5)*sin(x))
cos(x)^4/((cos(x)^5 + sin(x)^5)*sin(x))
Potencias [src] 
                                    4                   
                      / I*x    -I*x\                    
                      |e      e    |                    
                  2*I*|---- + -----|                    
                      \ 2       2  /                    
--------------------------------------------------------
/              5                     5\                 
|/ I*x    -I*x\      /   -I*x    I*x\ |                 
||e      e    |    I*\- e     + e   / | /   -I*x    I*x\
||---- + -----|  - -------------------|*\- e     + e   /
\\ 2       2  /             32        /
2i(eix2+e−ix2)4((eix2+e−ix2)5−i(eix−e−ix)532)(eix−e−ix)\frac{2 i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{4}}{\left(\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{5} - \frac{i \left(e^{i x} - e^{- i x}\right)^{5}}{32}\right) \left(e^{i x} - e^{- i x}\right)} 
2*i*(exp(i*x)/2 + exp(-i*x)/2)^4/(((exp(i*x)/2 + exp(-i*x)/2)^5 - i*(-exp(-i*x) + exp(i*x))^5/32)*(-exp(-i*x) + exp(i*x)))
Combinatoria [src] 
                                               4                                                
                                            cos (x)                                             
------------------------------------------------------------------------------------------------
                  /   4         4         2       2         3                3          \       
(cos(x) + sin(x))*\cos (x) + sin (x) + cos (x)*sin (x) - cos (x)*sin(x) - sin (x)*cos(x)/*sin(x)
cos⁡4(x)(sin⁡(x)+cos⁡(x))(sin⁡4(x)−sin⁡3(x)cos⁡(x)+sin⁡2(x)cos⁡2(x)−sin⁡(x)cos⁡3(x)+cos⁡4(x))sin⁡(x)\frac{\cos^{4}{\left(x \right)}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\sin^{4}{\left(x \right)} - \sin^{3}{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - \sin{\left(x \right)} \cos^{3}{\left(x \right)} + \cos^{4}{\left(x \right)}\right) \sin{\left(x \right)}} 
cos(x)^4/((cos(x) + sin(x))*(cos(x)^4 + sin(x)^4 + cos(x)^2*sin(x)^2 - cos(x)^3*sin(x) - sin(x)^3*cos(x))*sin(x))
Denominador común [src] 
           4            
        cos (x)         
------------------------
   6         5          
sin (x) + cos (x)*sin(x)
cos⁡4(x)sin⁡6(x)+sin⁡(x)cos⁡5(x)\frac{\cos^{4}{\left(x \right)}}{\sin^{6}{\left(x \right)} + \sin{\left(x \right)} \cos^{5}{\left(x \right)}} 
cos(x)^4/(sin(x)^6 + cos(x)^5*sin(x))
Abrimos la expresión [src] 
           4            
        cos (x)         
------------------------
   6         5          
sin (x) + cos (x)*sin(x)
cos⁡4(x)sin⁡6(x)+sin⁡(x)cos⁡5(x)\frac{\cos^{4}{\left(x \right)}}{\sin^{6}{\left(x \right)} + \sin{\left(x \right)} \cos^{5}{\left(x \right)}} 
cos(x)^4/(sin(x)^6 + cos(x)^5*sin(x))
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