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Simplificación de expresiones/ Descomposición en fracciones simples/ tgx+(cos^5x-sin^5x)/((1+sinx*cosx)*cos^2x)

¿Cómo vas a descomponer esta tgx+(cos^5x-sin^5x)/((1+sinx*cosx)*cos^2x) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                 5         5        
              cos (x) - sin (x)     
tan(x) + ---------------------------
                                2   
         (1 + sin(x)*cos(x))*cos (x)
$$\tan{\left(x \right)} + \frac{- \sin^{5}{\left(x \right)} + \cos^{5}{\left(x \right)}}{\left(\sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}$$ 
tan(x) + (cos(x)^5 - sin(x)^5)/(((1 + sin(x)*cos(x))*cos(x)^2))
Simplificación general [src] 
1      5      sin(2*x)      5      cos(4*x)
- + cos (x) + -------- - sin (x) - --------
8                2                    8    
-------------------------------------------
           /    sin(2*x)\    2             
           |1 + --------|*cos (x)          
           \       2    /
$$\frac{- \sin^{5}{\left(x \right)} + \frac{\sin{\left(2 x \right)}}{2} + \cos^{5}{\left(x \right)} - \frac{\cos{\left(4 x \right)}}{8} + \frac{1}{8}}{\left(\frac{\sin{\left(2 x \right)}}{2} + 1\right) \cos^{2}{\left(x \right)}}$$ 
(1/8 + cos(x)^5 + sin(2*x)/2 - sin(x)^5 - cos(4*x)/8)/((1 + sin(2*x)/2)*cos(x)^2)
Respuesta numérica [src] 
(cos(x)^5 - sin(x)^5)/((1.0 + cos(x)*sin(x))*cos(x)^2) + tan(x)
(cos(x)^5 - sin(x)^5)/((1.0 + cos(x)*sin(x))*cos(x)^2) + tan(x)
Unión de expresiones racionales [src] 
   5         5         2                              
cos (x) - sin (x) + cos (x)*(1 + cos(x)*sin(x))*tan(x)
------------------------------------------------------
                                    2                 
             (1 + cos(x)*sin(x))*cos (x)
$$\frac{\left(\sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} \tan{\left(x \right)} - \sin^{5}{\left(x \right)} + \cos^{5}{\left(x \right)}}{\left(\sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}$$ 
(cos(x)^5 - sin(x)^5 + cos(x)^2*(1 + cos(x)*sin(x))*tan(x))/((1 + cos(x)*sin(x))*cos(x)^2)
Potencias [src] 
                                            5                     5         
                              / I*x    -I*x\      /   -I*x    I*x\          
                              |e      e    |    I*\- e     + e   /          
  /   I*x    -I*x\            |---- + -----|  + -------------------         
I*\- e    + e    /            \ 2       2  /             32                 
------------------ + -------------------------------------------------------
    I*x    -I*x      /      / I*x    -I*x\                 \                
   e    + e          |      |e      e    | /   -I*x    I*x\|               2
                     |    I*|---- + -----|*\- e     + e   /| / I*x    -I*x\ 
                     |      \ 2       2  /                 | |e      e    | 
                     |1 - ---------------------------------|*|---- + -----| 
                     \                    2                / \ 2       2  /
$$\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{5} + \frac{i \left(e^{i x} - e^{- i x}\right)^{5}}{32}}{\left(- \frac{i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)}{2} + 1\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}}$$ 
i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)) + ((exp(i*x)/2 + exp(-i*x)/2)^5 + i*(-exp(-i*x) + exp(i*x))^5/32)/((1 - i*(exp(i*x)/2 + exp(-i*x)/2)*(-exp(-i*x) + exp(i*x))/2)*(exp(i*x)/2 + exp(-i*x)/2)^2)
Denominador común [src] 
      5         5                
   cos (x) - sin (x)             
------------------------ + tan(x)
   2         3                   
cos (x) + cos (x)*sin(x)
$$\tan{\left(x \right)} + \frac{- \sin^{5}{\left(x \right)} + \cos^{5}{\left(x \right)}}{\sin{\left(x \right)} \cos^{3}{\left(x \right)} + \cos^{2}{\left(x \right)}}$$ 
(cos(x)^5 - sin(x)^5)/(cos(x)^2 + cos(x)^3*sin(x)) + tan(x)
Combinatoria [src] 
   5         5         2                3                 
cos (x) - sin (x) + cos (x)*tan(x) + cos (x)*sin(x)*tan(x)
----------------------------------------------------------
                                      2                   
               (1 + cos(x)*sin(x))*cos (x)
$$\frac{- \sin^{5}{\left(x \right)} + \sin{\left(x \right)} \cos^{3}{\left(x \right)} \tan{\left(x \right)} + \cos^{5}{\left(x \right)} + \cos^{2}{\left(x \right)} \tan{\left(x \right)}}{\left(\sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}$$ 
(cos(x)^5 - sin(x)^5 + cos(x)^2*tan(x) + cos(x)^3*sin(x)*tan(x))/((1 + cos(x)*sin(x))*cos(x)^2)
Denominador racional [src] 
   5         5         2                              
cos (x) - sin (x) + cos (x)*(1 + cos(x)*sin(x))*tan(x)
------------------------------------------------------
                                    2                 
             (1 + cos(x)*sin(x))*cos (x)
$$\frac{\left(\sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)} \tan{\left(x \right)} - \sin^{5}{\left(x \right)} + \cos^{5}{\left(x \right)}}{\left(\sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}$$ 
(cos(x)^5 - sin(x)^5 + cos(x)^2*(1 + cos(x)*sin(x))*tan(x))/((1 + cos(x)*sin(x))*cos(x)^2)
Abrimos la expresión [src] 
           5                          5                     
        cos (x)                    sin (x)                  
------------------------ - ------------------------ + tan(x)
   2         3                2         3                   
cos (x) + cos (x)*sin(x)   cos (x) + cos (x)*sin(x)
$$\tan{\left(x \right)} - \frac{\sin^{5}{\left(x \right)}}{\sin{\left(x \right)} \cos^{3}{\left(x \right)} + \cos^{2}{\left(x \right)}} + \frac{\cos^{5}{\left(x \right)}}{\sin{\left(x \right)} \cos^{3}{\left(x \right)} + \cos^{2}{\left(x \right)}}$$ 
cos(x)^5/(cos(x)^2 + cos(x)^3*sin(x)) - sin(x)^5/(cos(x)^2 + cos(x)^3*sin(x)) + tan(x)
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