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Simplificación de expresiones/ Descomposición en fracciones simples/ tan(x)^2*(3+3*tan(x)^2)/(1+tan(x)^2)-tan(x)^4*(2+2*tan(x)^2)/(1+tan(x)^2)^2

¿Cómo vas a descomponer esta tan(x)^2*(3+3*tan(x)^2)/(1+tan(x)^2)-tan(x)^4*(2+2*tan(x)^2)/(1+tan(x)^2)^2 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   2    /         2   \      4    /         2   \
tan (x)*\3 + 3*tan (x)/   tan (x)*\2 + 2*tan (x)/
----------------------- - -----------------------
             2                              2    
      1 + tan (x)              /       2   \     
                               \1 + tan (x)/
$$- \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan^{4}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}} + \frac{\left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$ 
(tan(x)^2*(3 + 3*tan(x)^2))/(1 + tan(x)^2) - tan(x)^4*(2 + 2*tan(x)^2)/(1 + tan(x)^2)^2
Simplificación general [src] 
   2       2    /       2   \
cos (x)*tan (x)*\3 + tan (x)/
$$\left(\tan^{2}{\left(x \right)} + 3\right) \cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}$$ 
cos(x)^2*tan(x)^2*(3 + tan(x)^2)
Descomposición de una fracción [src] 
2 + tan(x)^2 - 2/(1 + tan(x)^2)
$$\tan^{2}{\left(x \right)} + 2 - \frac{2}{\tan^{2}{\left(x \right)} + 1}$$ 
       2           2     
2 + tan (x) - -----------
                     2   
              1 + tan (x)
Respuesta numérica [src] 
tan(x)^2*(3.0 + 3.0*tan(x)^2)/(1.0 + tan(x)^2) - tan(x)^4*(2.0 + 2.0*tan(x)^2)/(1.0 + tan(x)^2)^2
tan(x)^2*(3.0 + 3.0*tan(x)^2)/(1.0 + tan(x)^2) - tan(x)^4*(2.0 + 2.0*tan(x)^2)/(1.0 + tan(x)^2)^2
Denominador racional [src] 
             2                                                                
/       2   \     2    /         2   \      4    /       2   \ /         2   \
\1 + tan (x)/ *tan (x)*\3 + 3*tan (x)/ - tan (x)*\1 + tan (x)/*\2 + 2*tan (x)/
------------------------------------------------------------------------------
                                             3                                
                                /       2   \                                 
                                \1 + tan (x)/
$$\frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)} - \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan^{4}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{3}}$$ 
((1 + tan(x)^2)^2*tan(x)^2*(3 + 3*tan(x)^2) - tan(x)^4*(1 + tan(x)^2)*(2 + 2*tan(x)^2))/(1 + tan(x)^2)^3
Denominador común [src] 
       2           2     
2 + tan (x) - -----------
                     2   
              1 + tan (x)
$$\tan^{2}{\left(x \right)} + 2 - \frac{2}{\tan^{2}{\left(x \right)} + 1}$$ 
2 + tan(x)^2 - 2/(1 + tan(x)^2)
Combinatoria [src] 
   2    /       2   \
tan (x)*\3 + tan (x)/
---------------------
            2        
     1 + tan (x)
$$\frac{\left(\tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$ 
tan(x)^2*(3 + tan(x)^2)/(1 + tan(x)^2)
Compilar la expresión [src] 
   2    /         2   \      4    /         2   \
tan (x)*\3 + 3*tan (x)/   tan (x)*\2 + 2*tan (x)/
----------------------- - -----------------------
             2                              2    
      1 + tan (x)              /       2   \     
                               \1 + tan (x)/
$$\frac{\left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} - \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan^{4}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}$$ 
tan(x)^2*(3 + 3*tan(x)^2)/(1 + tan(x)^2) - tan(x)^4*(2 + 2*tan(x)^2)/(1 + tan(x)^2)^2
Abrimos la expresión [src] 
              4                         6                   2             4    
         2*tan (x)                 2*tan (x)           3*tan (x)     3*tan (x) 
- ----------------------- - ----------------------- + ----------- + -----------
         4           2             4           2             2             2   
  1 + tan (x) + 2*tan (x)   1 + tan (x) + 2*tan (x)   1 + tan (x)   1 + tan (x)
$$- \frac{2 \tan^{6}{\left(x \right)}}{\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1} - \frac{2 \tan^{4}{\left(x \right)}}{\tan^{4}{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + 1} + \frac{3 \tan^{4}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + \frac{3 \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$ 
-2*tan(x)^4/(1 + tan(x)^4 + 2*tan(x)^2) - 2*tan(x)^6/(1 + tan(x)^4 + 2*tan(x)^2) + 3*tan(x)^2/(1 + tan(x)^2) + 3*tan(x)^4/(1 + tan(x)^2)
Potencias [src] 
   2    /         2   \      4    /          2   \
tan (x)*\3 + 3*tan (x)/   tan (x)*\-2 - 2*tan (x)/
----------------------- + ------------------------
             2                              2     
      1 + tan (x)              /       2   \      
                               \1 + tan (x)/
$$\frac{\left(- 2 \tan^{2}{\left(x \right)} - 2\right) \tan^{4}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}} + \frac{\left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$ 
   2    /         2   \      4    /         2   \
tan (x)*\3 + 3*tan (x)/   tan (x)*\2 + 2*tan (x)/
----------------------- - -----------------------
             2                              2    
      1 + tan (x)              /       2   \     
                               \1 + tan (x)/
$$\frac{\left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} - \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan^{4}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}$$ 
                    /                      2\                     /                      2\
                  2 |      /   I*x    -I*x\ |                   4 |      /   I*x    -I*x\ |
  /   I*x    -I*x\  |    3*\- e    + e    / |   /   I*x    -I*x\  |    2*\- e    + e    / |
  \- e    + e    / *|3 - -------------------|   \- e    + e    / *|2 - -------------------|
                    |                    2  |                     |                    2  |
                    |      / I*x    -I*x\   |                     |      / I*x    -I*x\   |
                    \      \e    + e    /   /                     \      \e    + e    /   /
- ------------------------------------------- - -------------------------------------------
    /                    2\                                              2                 
    |    /   I*x    -I*x\ |               2       /                    2\                  
    |    \- e    + e    / | / I*x    -I*x\        |    /   I*x    -I*x\ |                4 
    |1 - -----------------|*\e    + e    /        |    \- e    + e    / |  / I*x    -I*x\  
    |                   2 |                       |1 - -----------------| *\e    + e    /  
    |     / I*x    -I*x\  |                       |                   2 |                  
    \     \e    + e    /  /                       |     / I*x    -I*x\  |                  
                                                  \     \e    + e    /  /
$$- \frac{\left(- \frac{3 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 3\right) \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1\right) \left(e^{i x} + e^{- i x}\right)^{2}} - \frac{\left(- \frac{2 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 2\right) \left(- e^{i x} + e^{- i x}\right)^{4}}{\left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1\right)^{2} \left(e^{i x} + e^{- i x}\right)^{4}}$$ 
-(-exp(i*x) + exp(-i*x))^2*(3 - 3*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)/((1 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x))^2) - (-exp(i*x) + exp(-i*x))^4*(2 - 2*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)/((1 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)^2*(exp(i*x) + exp(-i*x))^4)
Unión de expresiones racionales [src] 
   2    /       2   \
tan (x)*\3 + tan (x)/
---------------------
            2        
     1 + tan (x)
$$\frac{\left(\tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$ 
tan(x)^2*(3 + tan(x)^2)/(1 + tan(x)^2)
Parte trigonométrica [src] 
           /          2     \                /          2     \   
      2    |     3*sec (x)  |           4    |     2*sec (x)  |   
   sec (x)*|3 + ------------|        sec (x)*|2 + ------------|   
           |       2/    pi\|                |       2/    pi\|   
           |    sec |x - --||                |    sec |x - --||   
           \        \    2 //                \        \    2 //   
------------------------------- - --------------------------------
/         2      \                                  2             
|      sec (x)   |    2/    pi\   /         2      \              
|1 + ------------|*sec |x - --|   |      sec (x)   |     4/    pi\
|       2/    pi\|     \    2 /   |1 + ------------| *sec |x - --|
|    sec |x - --||                |       2/    pi\|      \    2 /
\        \    2 //                |    sec |x - --||              
                                  \        \    2 //
$$\frac{\left(\frac{3 \sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 3\right) \sec^{2}{\left(x \right)}}{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{\left(\frac{2 \sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 2\right) \sec^{4}{\left(x \right)}}{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{4}{\left(x - \frac{\pi}{2} \right)}}$$ 
                 4   
     2      4*sin (x)
2*sin (x) + ---------
               2     
            sin (2*x)
$$\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 2 \sin^{2}{\left(x \right)}$$ 
                    2/    pi\
                 cos |x - --|
     2/    pi\       \    2 /
2*cos |x - --| + ------------
      \    2 /        2      
                   cos (x)
$$2 \cos^{2}{\left(x - \frac{\pi}{2} \right)} + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}$$ 
        /         2   \           /         2   \
   2    |    3*sin (x)|      4    |    2*sin (x)|
sin (x)*|3 + ---------|   sin (x)*|2 + ---------|
        |        2    |           |        2    |
        \     cos (x) /           \     cos (x) /
----------------------- - -----------------------
 /       2   \                          2        
 |    sin (x)|    2        /       2   \         
 |1 + -------|*cos (x)     |    sin (x)|     4   
 |       2   |             |1 + -------| *cos (x)
 \    cos (x)/             |       2   |         
                           \    cos (x)/
$$\frac{\left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 3\right) \sin^{2}{\left(x \right)}}{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos^{2}{\left(x \right)}} - \frac{\left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \sin^{4}{\left(x \right)}}{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)^{2} \cos^{4}{\left(x \right)}}$$ 
   2    /         2   \      4    /         2   \
tan (x)*\3 + 3*tan (x)/   tan (x)*\2 + 2*tan (x)/
----------------------- - -----------------------
             2                              2    
      1 + tan (x)              /       2   \     
                               \1 + tan (x)/
$$\frac{\left(3 \tan^{2}{\left(x \right)} + 3\right) \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} - \frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan^{4}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}$$ 
                 2/x\   
            8*cot |-|   
   1              \2/   
------- + --------------
   2                   2
cot (x)   /       2/x\\ 
          |1 + cot |-|| 
          \        \2//
$$\frac{1}{\cot^{2}{\left(x \right)}} + \frac{8 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$ 
             2/pi    \
          csc |-- - x|
   2          \2     /
------- + ------------
   2           2      
csc (x)     csc (x)
$$\frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}} + \frac{2}{\csc^{2}{\left(x \right)}}$$ 
             /         2/pi    \\                /         2/pi    \\
             |    3*csc |-- - x||                |    2*csc |-- - x||
   2/pi    \ |          \2     /|      4/pi    \ |          \2     /|
csc |-- - x|*|3 + --------------|   csc |-- - x|*|2 + --------------|
    \2     / |          2       |       \2     / |          2       |
             \       csc (x)    /                \       csc (x)    /
--------------------------------- - ---------------------------------
    /       2/pi    \\                                   2           
    |    csc |-- - x||                 /       2/pi    \\            
    |        \2     /|    2            |    csc |-- - x||            
    |1 + ------------|*csc (x)         |        \2     /|     4      
    |         2      |                 |1 + ------------| *csc (x)   
    \      csc (x)   /                 |         2      |            
                                       \      csc (x)   /
$$\frac{\left(3 + \frac{3 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc^{2}{\left(x \right)}} - \frac{\left(2 + \frac{2 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc^{4}{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right)^{2} \csc^{4}{\left(x \right)}}$$ 
                    2      
     2           sec (x)   
------------ + ------------
   2/    pi\      2/    pi\
sec |x - --|   sec |x - --|
    \    2 /       \    2 /
$$\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{2}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}$$ 
            3                       2         
     3 + -------             2 + -------      
            2                       2         
         cot (x)                 cot (x)      
--------------------- - ----------------------
/       1   \    2                   2        
|1 + -------|*cot (x)   /       1   \     4   
|       2   |           |1 + -------| *cot (x)
\    cot (x)/           |       2   |         
                        \    cot (x)/
$$\frac{3 + \frac{3}{\cot^{2}{\left(x \right)}}}{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot^{2}{\left(x \right)}} - \frac{2 + \frac{2}{\cot^{2}{\left(x \right)}}}{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right)^{2} \cot^{4}{\left(x \right)}}$$ 
             /         4   \             /          4   \
        8    |    8*sin (x)|        4    |    12*sin (x)|
  16*sin (x)*|2 + ---------|   4*sin (x)*|3 + ----------|
             |       2     |             |       2      |
             \    sin (2*x)/             \    sin (2*x) /
- -------------------------- + --------------------------
                 2             /         4   \           
  /         4   \              |    4*sin (x)|    2      
  |    4*sin (x)|     4        |1 + ---------|*sin (2*x) 
  |1 + ---------| *sin (2*x)   |       2     |           
  |       2     |              \    sin (2*x)/           
  \    sin (2*x)/
$$\frac{4 \left(\frac{12 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 3\right) \sin^{4}{\left(x \right)}}{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin^{2}{\left(2 x \right)}} - \frac{16 \left(\frac{8 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 2\right) \sin^{8}{\left(x \right)}}{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right)^{2} \sin^{4}{\left(2 x \right)}}$$ 
                 2/x\   
            8*tan |-|   
   2              \2/   
tan (x) + --------------
                       2
          /       2/x\\ 
          |1 + tan |-|| 
          \        \2//
$$\tan^{2}{\left(x \right)} + \frac{8 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$ 
        /         2   \           /         2   \
   2    |    3*sec (x)|      4    |    2*sec (x)|
sec (x)*|3 + ---------|   sec (x)*|2 + ---------|
        |        2    |           |        2    |
        \     csc (x) /           \     csc (x) /
----------------------- - -----------------------
 /       2   \                          2        
 |    sec (x)|    2        /       2   \         
 |1 + -------|*csc (x)     |    sec (x)|     4   
 |       2   |             |1 + -------| *csc (x)
 \    csc (x)/             |       2   |         
                           \    csc (x)/
$$\frac{\left(3 + \frac{3 \sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \sec^{2}{\left(x \right)}}{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \csc^{2}{\left(x \right)}} - \frac{\left(2 + \frac{2 \sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \sec^{4}{\left(x \right)}}{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right)^{2} \csc^{4}{\left(x \right)}}$$ 
   2           2   
tan (x) + 2*sin (x)
$$2 \sin^{2}{\left(x \right)} + \tan^{2}{\left(x \right)}$$ 
             /         2/    pi\\                /         2/    pi\\
             |    3*cos |x - --||                |    2*cos |x - --||
   2/    pi\ |          \    2 /|      4/    pi\ |          \    2 /|
cos |x - --|*|3 + --------------|   cos |x - --|*|2 + --------------|
    \    2 / |          2       |       \    2 / |          2       |
             \       cos (x)    /                \       cos (x)    /
--------------------------------- - ---------------------------------
    /       2/    pi\\                                   2           
    |    cos |x - --||                 /       2/    pi\\            
    |        \    2 /|    2            |    cos |x - --||            
    |1 + ------------|*cos (x)         |        \    2 /|     4      
    |         2      |                 |1 + ------------| *cos (x)   
    \      cos (x)   /                 |         2      |            
                                       \      cos (x)   /
$$\frac{\left(3 + \frac{3 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)}} - \frac{\left(2 + \frac{2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos^{4}{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right)^{2} \cos^{4}{\left(x \right)}}$$ 
cos(x - pi/2)^2*(3 + 3*cos(x - pi/2)^2/cos(x)^2)/((1 + cos(x - pi/2)^2/cos(x)^2)*cos(x)^2) - cos(x - pi/2)^4*(2 + 2*cos(x - pi/2)^2/cos(x)^2)/((1 + cos(x - pi/2)^2/cos(x)^2)^2*cos(x)^4)
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