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Simplificación de expresiones/ Descomposición en fracciones simples/ tgx+(cos^3x-sin^3x)/(1+sinx*cosx)*cos^2x

¿Cómo vas a descomponer esta tgx+(cos^3x-sin^3x)/(1+sinx*cosx)*cos^2x expresión en fracciones?

Expresión a simplificar:

Solución

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