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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(x)/(x+1)^3-3*sin(x)/(x+1)^4

¿Cómo vas a descomponer esta cos(x)/(x+1)^3-3*sin(x)/(x+1)^4 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
 cos(x)    3*sin(x)
-------- - --------
       3          4
(x + 1)    (x + 1)
$$- \frac{3 \sin{\left(x \right)}}{\left(x + 1\right)^{4}} + \frac{\cos{\left(x \right)}}{\left(x + 1\right)^{3}}$$ 
cos(x)/(x + 1)^3 - 3*sin(x)/(x + 1)^4
Simplificación general [src] 
-3*sin(x) + (1 + x)*cos(x)
--------------------------
                4         
         (1 + x)
$$\frac{\left(x + 1\right) \cos{\left(x \right)} - 3 \sin{\left(x \right)}}{\left(x + 1\right)^{4}}$$ 
(-3*sin(x) + (1 + x)*cos(x))/(1 + x)^4
Respuesta numérica [src] 
cos(x)/(1.0 + x)^3 - 3.0*sin(x)/(1.0 + x)^4
cos(x)/(1.0 + x)^3 - 3.0*sin(x)/(1.0 + x)^4
Denominador común [src] 
-3*sin(x) + x*cos(x) + cos(x)
-----------------------------
       4            3      2 
  1 + x  + 4*x + 4*x  + 6*x
$$\frac{x \cos{\left(x \right)} - 3 \sin{\left(x \right)} + \cos{\left(x \right)}}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1}$$ 
(-3*sin(x) + x*cos(x) + cos(x))/(1 + x^4 + 4*x + 4*x^3 + 6*x^2)
Potencias [src] 
 I*x    -I*x                       
e      e                           
---- + -----       /   -I*x    I*x\
 2       2     3*I*\- e     + e   /
------------ + --------------------
         3                   4     
  (1 + x)           2*(1 + x)
$$\frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\left(x + 1\right)^{3}} + \frac{3 i \left(e^{i x} - e^{- i x}\right)}{2 \left(x + 1\right)^{4}}$$ 
 cos(x)    3*sin(x)
-------- - --------
       3          4
(1 + x)    (1 + x)
$$\frac{\cos{\left(x \right)}}{\left(x + 1\right)^{3}} - \frac{3 \sin{\left(x \right)}}{\left(x + 1\right)^{4}}$$ 
cos(x)/(1 + x)^3 - 3*sin(x)/(1 + x)^4
Denominador racional [src] 
       4                   3       
(1 + x) *cos(x) - 3*(1 + x) *sin(x)
-----------------------------------
                     7             
              (1 + x)
$$\frac{\left(x + 1\right)^{4} \cos{\left(x \right)} - 3 \left(x + 1\right)^{3} \sin{\left(x \right)}}{\left(x + 1\right)^{7}}$$ 
((1 + x)^4*cos(x) - 3*(1 + x)^3*sin(x))/(1 + x)^7
Unión de expresiones racionales [src] 
-3*sin(x) + (1 + x)*cos(x)
--------------------------
                4         
         (1 + x)
$$\frac{\left(x + 1\right) \cos{\left(x \right)} - 3 \sin{\left(x \right)}}{\left(x + 1\right)^{4}}$$ 
(-3*sin(x) + (1 + x)*cos(x))/(1 + x)^4
Compilar la expresión [src] 
 cos(x)    3*sin(x)
-------- - --------
       3          4
(1 + x)    (1 + x)
$$\frac{\cos{\left(x \right)}}{\left(x + 1\right)^{3}} - \frac{3 \sin{\left(x \right)}}{\left(x + 1\right)^{4}}$$ 
cos(x)/(1 + x)^3 - 3*sin(x)/(1 + x)^4
Parte trigonométrica [src] 
   /    pi\           
sin|x + --|           
   \    2 /   3*sin(x)
----------- - --------
         3           4
  (1 + x)     (1 + x)
$$\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\left(x + 1\right)^{3}} - \frac{3 \sin{\left(x \right)}}{\left(x + 1\right)^{4}}$$ 
       1                   3          
--------------- - --------------------
       3                 4    /    pi\
(1 + x) *sec(x)   (1 + x) *sec|x - --|
                              \    2 /
$$\frac{1}{\left(x + 1\right)^{3} \sec{\left(x \right)}} - \frac{3}{\left(x + 1\right)^{4} \sec{\left(x - \frac{\pi}{2} \right)}}$$ 
 cos(x)    3*sin(x)
-------- - --------
       3          4
(x + 1)    (x + 1)
$$\frac{\cos{\left(x \right)}}{\left(x + 1\right)^{3}} - \frac{3 \sin{\left(x \right)}}{\left(x + 1\right)^{4}}$$ 
       1                 3       
--------------- - ---------------
       3                 4       
(1 + x) *sec(x)   (1 + x) *csc(x)
$$\frac{1}{\left(x + 1\right)^{3} \sec{\left(x \right)}} - \frac{3}{\left(x + 1\right)^{4} \csc{\left(x \right)}}$$ 
         1                    3       
-------------------- - ---------------
       3    /pi    \          4       
(1 + x) *csc|-- - x|   (1 + x) *csc(x)
            \2     /
$$\frac{1}{\left(x + 1\right)^{3} \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{3}{\left(x + 1\right)^{4} \csc{\left(x \right)}}$$ 
            2/x\                     /x\       
     1 - tan |-|                6*tan|-|       
             \2/                     \2/       
---------------------- - ----------------------
       3 /       2/x\\          4 /       2/x\\
(1 + x) *|1 + tan |-||   (1 + x) *|1 + tan |-||
         \        \2//            \        \2//
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(x + 1\right)^{3} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{6 \tan{\left(\frac{x}{2} \right)}}{\left(x + 1\right)^{4} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
 cos(x)    3*sin(x)
-------- - --------
       3          4
(1 + x)    (1 + x)
$$\frac{\cos{\left(x \right)}}{\left(x + 1\right)^{3}} - \frac{3 \sin{\left(x \right)}}{\left(x + 1\right)^{4}}$$ 
             2/x\                    /x\       
     -1 + cot |-|               6*cot|-|       
              \2/                    \2/       
---------------------- - ----------------------
       3 /       2/x\\          4 /       2/x\\
(1 + x) *|1 + cot |-||   (1 + x) *|1 + cot |-||
         \        \2//            \        \2//
$$\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(x + 1\right)^{3} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{6 \cot{\left(\frac{x}{2} \right)}}{\left(x + 1\right)^{4} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$ 
                /    pi\
           3*cos|x - --|
 cos(x)         \    2 /
-------- - -------------
       3             4  
(1 + x)       (1 + x)
$$\frac{\cos{\left(x \right)}}{\left(x + 1\right)^{3}} - \frac{3 \cos{\left(x - \frac{\pi}{2} \right)}}{\left(x + 1\right)^{4}}$$ 
cos(x)/(1 + x)^3 - 3*cos(x - pi/2)/(1 + x)^4
Combinatoria [src] 
-3*sin(x) + x*cos(x) + cos(x)
-----------------------------
                  4          
           (1 + x)
$$\frac{x \cos{\left(x \right)} - 3 \sin{\left(x \right)} + \cos{\left(x \right)}}{\left(x + 1\right)^{4}}$$ 
(-3*sin(x) + x*cos(x) + cos(x))/(1 + x)^4
Abrimos la expresión [src] 
       cos(x)                  3*sin(x)         
------------------- - --------------------------
     3            2        4            3      2
1 + x  + 3*x + 3*x    1 + x  + 4*x + 4*x  + 6*x
$$- \frac{3 \sin{\left(x \right)}}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1} + \frac{\cos{\left(x \right)}}{x^{3} + 3 x^{2} + 3 x + 1}$$ 
cos(x)/(1 + x^3 + 3*x + 3*x^2) - 3*sin(x)/(1 + x^4 + 4*x + 4*x^3 + 6*x^2)
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