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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    tan(x)               3*x
------------- + tan(x) - ---
     2                    2 
2*tan (x) + 2
$$- \frac{3 x}{2} + \left(\tan{\left(x \right)} + \frac{\tan{\left(x \right)}}{2 \tan^{2}{\left(x \right)} + 2}\right)$$ 
tan(x)/(2*tan(x)^2 + 2) + tan(x) - 3*x/2
Simplificación general [src] 
  3*x   sin(2*x)         
- --- + -------- + tan(x)
   2       4
$$- \frac{3 x}{2} + \frac{\sin{\left(2 x \right)}}{4} + \tan{\left(x \right)}$$ 
-3*x/2 + sin(2*x)/4 + tan(x)
Respuesta numérica [src] 
-1.5*x + tan(x)/(2.0 + 2.0*tan(x)^2) + tan(x)
-1.5*x + tan(x)/(2.0 + 2.0*tan(x)^2) + tan(x)
Denominador común [src] 
  3*x       tan(x)            
- --- + ------------- + tan(x)
   2             2            
        2 + 2*tan (x)
$$- \frac{3 x}{2} + \tan{\left(x \right)} + \frac{\tan{\left(x \right)}}{2 \tan^{2}{\left(x \right)} + 2}$$ 
-3*x/2 + tan(x)/(2 + 2*tan(x)^2) + tan(x)
Potencias [src] 
  3*x       tan(x)            
- --- + ------------- + tan(x)
   2             2            
        2 + 2*tan (x)
$$- \frac{3 x}{2} + \tan{\left(x \right)} + \frac{\tan{\left(x \right)}}{2 \tan^{2}{\left(x \right)} + 2}$$ 
          /   I*x    -I*x\                /   I*x    -I*x\           
  3*x   I*\- e    + e    /              I*\- e    + e    /           
- --- + ------------------ + ----------------------------------------
   2        I*x    -I*x      /                      2\               
           e    + e          |      /   I*x    -I*x\ |               
                             |    2*\- e    + e    / | / I*x    -I*x\
                             |2 - -------------------|*\e    + e    /
                             |                    2  |               
                             |      / I*x    -I*x\   |               
                             \      \e    + e    /   /
$$- \frac{3 x}{2} + \frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + \frac{i \left(- e^{i x} + e^{- i x}\right)}{\left(- \frac{2 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 2\right) \left(e^{i x} + e^{- i x}\right)}$$ 
-3*x/2 + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)) + i*(-exp(i*x) + exp(-i*x))/((2 - 2*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x)))
Combinatoria [src] 
 /                 3                   2   \ 
-\-3*tan(x) - 2*tan (x) + 3*x + 3*x*tan (x)/ 
---------------------------------------------
                 /       2   \               
               2*\1 + tan (x)/
$$- \frac{3 x \tan^{2}{\left(x \right)} + 3 x - 2 \tan^{3}{\left(x \right)} - 3 \tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$ 
-(-3*tan(x) - 2*tan(x)^3 + 3*x + 3*x*tan(x)^2)/(2*(1 + tan(x)^2))
Compilar la expresión [src] 
  3*x       tan(x)            
- --- + ------------- + tan(x)
   2             2            
        2 + 2*tan (x)
$$- \frac{3 x}{2} + \tan{\left(x \right)} + \frac{\tan{\left(x \right)}}{2 \tan^{2}{\left(x \right)} + 2}$$ 
-3*x/2 + tan(x)/(2 + 2*tan(x)^2) + tan(x)
Abrimos la expresión [src] 
  3*x       tan(x)            
- --- + ------------- + tan(x)
   2         2                
        2*tan (x) + 2
$$- \frac{3 x}{2} + \tan{\left(x \right)} + \frac{\tan{\left(x \right)}}{2 \tan^{2}{\left(x \right)} + 2}$$ 
-3*x/2 + tan(x)/(2*tan(x)^2 + 2) + tan(x)
Parte trigonométrica [src] 
             2                  2           
  3*x   2*sin (x)          2*sin (x)        
- --- + --------- + ------------------------
   2     sin(2*x)   /         4   \         
                    |    8*sin (x)|         
                    |2 + ---------|*sin(2*x)
                    |       2     |         
                    \    sin (2*x)/
$$- \frac{3 x}{2} + \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{2 \sin^{2}{\left(x \right)}}{\left(\frac{8 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 2\right) \sin{\left(2 x \right)}}$$ 
  3*x   sin(2*x)         
- --- + -------- + tan(x)
   2       4
$$- \frac{3 x}{2} + \frac{\sin{\left(2 x \right)}}{4} + \tan{\left(x \right)}$$ 
  3*x   sec(x)           sec(x)        
- --- + ------ + ----------------------
   2    csc(x)   /         2   \       
                 |    2*sec (x)|       
                 |2 + ---------|*csc(x)
                 |        2    |       
                 \     csc (x) /
$$- \frac{3 x}{2} + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} + \frac{\sec{\left(x \right)}}{\left(2 + \frac{2 \sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}$$ 
  3*x   sin(x)           sin(x)        
- --- + ------ + ----------------------
   2    cos(x)   /         2   \       
                 |    2*sin (x)|       
                 |2 + ---------|*cos(x)
                 |        2    |       
                 \     cos (x) /
$$- \frac{3 x}{2} + \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \cos{\left(x \right)}}$$ 
  3*x        tan(x)             
- --- + --------------- + tan(x)
   2      /       2   \         
        2*\1 + tan (x)/
$$- \frac{3 x}{2} + \tan{\left(x \right)} + \frac{\tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$ 
                        2   
  3*x   sin(2*x)   2*sin (x)
- --- + -------- + ---------
   2       4        sin(2*x)
$$- \frac{3 x}{2} + \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\sin{\left(2 x \right)}}{4}$$ 
  3*x          1             sec(x)  
- --- + --------------- + -----------
   2         /      pi\      /    pi\
        4*sec|2*x - --|   sec|x - --|
             \      2 /      \    2 /
$$- \frac{3 x}{2} + \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{4 \sec{\left(2 x - \frac{\pi}{2} \right)}}$$ 
  3*x      sec(x)                 sec(x)            
- --- + ----------- + ------------------------------
   2       /    pi\   /          2     \            
        sec|x - --|   |     2*sec (x)  |    /    pi\
           \    2 /   |2 + ------------|*sec|x - --|
                      |       2/    pi\|    \    2 /
                      |    sec |x - --||            
                      \        \    2 //
$$- \frac{3 x}{2} + \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\sec{\left(x \right)}}{\left(\frac{2 \sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 2\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$ 
           /pi    \              /pi    \        
        csc|-- - x|           csc|-- - x|        
  3*x      \2     /              \2     /        
- --- + ----------- + ---------------------------
   2       csc(x)     /         2/pi    \\       
                      |    2*csc |-- - x||       
                      |          \2     /|       
                      |2 + --------------|*csc(x)
                      |          2       |       
                      \       csc (x)    /
$$- \frac{3 x}{2} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\left(2 + \frac{2 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}$$ 
  1      3*x            1          
------ - --- + --------------------
cot(x)    2    /       2   \       
               |2 + -------|*cot(x)
               |       2   |       
               \    cot (x)/
$$- \frac{3 x}{2} + \frac{1}{\cot{\left(x \right)}} + \frac{1}{\left(2 + \frac{2}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}}$$ 
           /      pi\      /    pi\
        cos|2*x - --|   cos|x - --|
  3*x      \      2 /      \    2 /
- --- + ------------- + -----------
   2          4            cos(x)
$$- \frac{3 x}{2} + \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{4} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}$$ 
  3*x       tan(x)            
- --- + ------------- + tan(x)
   2             2            
        2 + 2*tan (x)
$$- \frac{3 x}{2} + \tan{\left(x \right)} + \frac{\tan{\left(x \right)}}{2 \tan^{2}{\left(x \right)} + 2}$$ 
           /    pi\              /    pi\        
        cos|x - --|           cos|x - --|        
  3*x      \    2 /              \    2 /        
- --- + ----------- + ---------------------------
   2       cos(x)     /         2/    pi\\       
                      |    2*cos |x - --||       
                      |          \    2 /|       
                      |2 + --------------|*cos(x)
                      |          2       |       
                      \       cos (x)    /
$$- \frac{3 x}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(2 + \frac{2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}$$ 
  1      3*x        cot(x)    
------ - --- + ---------------
cot(x)    2      /       2   \
               2*\1 + cot (x)/
$$- \frac{3 x}{2} + \frac{1}{\cot{\left(x \right)}} + \frac{\cot{\left(x \right)}}{2 \left(\cot^{2}{\left(x \right)} + 1\right)}$$ 
                        /pi    \
                     csc|-- - x|
  3*x       1           \2     /
- --- + ---------- + -----------
   2    4*csc(2*x)      csc(x)
$$- \frac{3 x}{2} + \frac{1}{4 \csc{\left(2 x \right)}} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}$$ 
-3*x/2 + 1/(4*csc(2*x)) + csc(pi/2 - x)/csc(x)
Unión de expresiones racionales [src] 
/         2   \              /       2   \
\3 + 2*tan (x)/*tan(x) - 3*x*\1 + tan (x)/
------------------------------------------
               /       2   \              
             2*\1 + tan (x)/
$$\frac{- 3 x \left(\tan^{2}{\left(x \right)} + 1\right) + \left(2 \tan^{2}{\left(x \right)} + 3\right) \tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$ 
((3 + 2*tan(x)^2)*tan(x) - 3*x*(1 + tan(x)^2))/(2*(1 + tan(x)^2))
Denominador racional [src] 
               /         2   \     /         2   \       
2*tan(x) - 3*x*\2 + 2*tan (x)/ + 2*\2 + 2*tan (x)/*tan(x)
---------------------------------------------------------
                               2                         
                      4 + 4*tan (x)
$$\frac{- 3 x \left(2 \tan^{2}{\left(x \right)} + 2\right) + 2 \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} + 2 \tan{\left(x \right)}}{4 \tan^{2}{\left(x \right)} + 4}$$ 
(2*tan(x) - 3*x*(2 + 2*tan(x)^2) + 2*(2 + 2*tan(x)^2)*tan(x))/(4 + 4*tan(x)^2)
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