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1/(x^4+1)
Factorizar el polinomio:
z^2-z
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x^2-9
Descomponer al cuadrado:
2*x^2+x*y-y^2
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x^2+2*x-1
-y^4-y^2-6
Expresiones idénticas
cot(x/ dos)*(tan(x/ dos)^ cuatro + dos *tan(x/ dos)^ dos + uno)/(dos *tan(x/ dos)^ dos + cuatro *tan(x/ dos)+ dos)
cotangente de (x dividir por 2) multiplicar por ( tangente de (x dividir por 2) en el grado 4 más 2 multiplicar por tangente de (x dividir por 2) al cuadrado más 1) dividir por (2 multiplicar por tangente de (x dividir por 2) al cuadrado más 4 multiplicar por tangente de (x dividir por 2) más 2)
cotangente de (x dividir por dos) multiplicar por ( tangente de (x dividir por dos) en el grado cuatro más dos multiplicar por tangente de (x dividir por dos) en el grado dos más uno) dividir por (dos multiplicar por tangente de (x dividir por dos) en el grado dos más cuatro multiplicar por tangente de (x dividir por dos) más dos)
cot(x/2)*(tan(x/2)4+2*tan(x/2)2+1)/(2*tan(x/2)2+4*tan(x/2)+2)
cotx/2*tanx/24+2*tanx/22+1/2*tanx/22+4*tanx/2+2
cot(x/2)*(tan(x/2)⁴+2*tan(x/2)²+1)/(2*tan(x/2)²+4*tan(x/2)+2)
cot(x/2)*(tan(x/2) en el grado 4+2*tan(x/2) en el grado 2+1)/(2*tan(x/2) en el grado 2+4*tan(x/2)+2)
cot(x/2)(tan(x/2)^4+2tan(x/2)^2+1)/(2tan(x/2)^2+4tan(x/2)+2)
cot(x/2)(tan(x/2)4+2tan(x/2)2+1)/(2tan(x/2)2+4tan(x/2)+2)
cotx/2tanx/24+2tanx/22+1/2tanx/22+4tanx/2+2
cotx/2tanx/2^4+2tanx/2^2+1/2tanx/2^2+4tanx/2+2
cot(x dividir por 2)*(tan(x dividir por 2)^4+2*tan(x dividir por 2)^2+1) dividir por (2*tan(x dividir por 2)^2+4*tan(x dividir por 2)+2)
Expresiones semejantes
cot(x/2)*(tan(x/2)^4+2*tan(x/2)^2+1)/(2*tan(x/2)^2-4*tan(x/2)+2)
cot(x/2)*(tan(x/2)^4+2*tan(x/2)^2+1)/(2*tan(x/2)^2+4*tan(x/2)-2)
cot(x/2)*(tan(x/2)^4+2*tan(x/2)^2-1)/(2*tan(x/2)^2+4*tan(x/2)+2)
cot(x/2)*(tan(x/2)^4-2*tan(x/2)^2+1)/(2*tan(x/2)^2+4*tan(x/2)+2)
Expresiones con funciones
Cotangente cot
cot(3*pi/(2+x))*cot(pi-x)-sin(pi+x)*cos(pi/(2+x))
cot(5*x)^2*(-15-15*cot(5*x)^2)*asin(sqrt(x))+cot(5*x)^3/(2*sqrt(x)*sqrt(1-x))
cot(pi/(2-a))*(-sin(a-3*pi/2)-sin(a))/((cot(a)*(cos(a)-(-1)*sin(a))))
cot(x)^atan(4*x)*(4*log(cot(x))/(1+16*x^2)+(-1-cot(x)^2)*atan(4*x)/cot(x))
cot(x)*(-2)-2*cot(x)^3/3-2*cos(x)/(3*sin(x))+2*cos(x)/((3*sin(x)^3))
Tangente tan
tan(x)^(cos(sqrt(x))^3)*(cos(sqrt(x))^3*(1+tan(x)^2)/tan(x)-3*cos(sqrt(x))^2*log(tan(x))*sin(sqrt(x))/(2*sqrt(x)))
tan(2*a)*tan(a)/((tan(2*a)*tan(a)))
tan(x+pi/8)/2-1/(2*tan(x+pi/8))
tan(E^asin(x))^3/(2*sqrt(x))+3*sqrt(x)*tan(E^asin(x))^2*(1+tan(E^asin(x))^2)*exp(asin(x))/sqrt(1-x^2)
tan2αc2α-(1-sin^25α)/(cos^25α-1)
Tangente tan
tan(x)^(cos(sqrt(x))^3)*(cos(sqrt(x))^3*(1+tan(x)^2)/tan(x)-3*cos(sqrt(x))^2*log(tan(x))*sin(sqrt(x))/(2*sqrt(x)))
tan(2*a)*tan(a)/((tan(2*a)*tan(a)))
tan(x+pi/8)/2-1/(2*tan(x+pi/8))
tan(E^asin(x))^3/(2*sqrt(x))+3*sqrt(x)*tan(E^asin(x))^2*(1+tan(E^asin(x))^2)*exp(asin(x))/sqrt(1-x^2)
tan2αc2α-(1-sin^25α)/(cos^25α-1)
Tangente tan
tan(x)^(cos(sqrt(x))^3)*(cos(sqrt(x))^3*(1+tan(x)^2)/tan(x)-3*cos(sqrt(x))^2*log(tan(x))*sin(sqrt(x))/(2*sqrt(x)))
tan(2*a)*tan(a)/((tan(2*a)*tan(a)))
tan(x+pi/8)/2-1/(2*tan(x+pi/8))
tan(E^asin(x))^3/(2*sqrt(x))+3*sqrt(x)*tan(E^asin(x))^2*(1+tan(E^asin(x))^2)*exp(asin(x))/sqrt(1-x^2)
tan2αc2α-(1-sin^25α)/(cos^25α-1)
Tangente tan
tan(x)^(cos(sqrt(x))^3)*(cos(sqrt(x))^3*(1+tan(x)^2)/tan(x)-3*cos(sqrt(x))^2*log(tan(x))*sin(sqrt(x))/(2*sqrt(x)))
tan(2*a)*tan(a)/((tan(2*a)*tan(a)))
tan(x+pi/8)/2-1/(2*tan(x+pi/8))
tan(E^asin(x))^3/(2*sqrt(x))+3*sqrt(x)*tan(E^asin(x))^2*(1+tan(E^asin(x))^2)*exp(asin(x))/sqrt(1-x^2)
tan2αc2α-(1-sin^25α)/(cos^25α-1)

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

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

Solución

Ha introducido [src]

   /x\ /   4/x\        2/x\    \
cot|-|*|tan |-| + 2*tan |-| + 1|
   \2/ \    \2/         \2/    /
--------------------------------
         2/x\        /x\        
    2*tan |-| + 4*tan|-| + 2    
          \2/        \2/

$$\frac{\left(\left(\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)}\right) + 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)}\right) + 2}$$

(cot(x/2)*(tan(x/2)^4 + 2*tan(x/2)^2 + 1))/(2*tan(x/2)^2 + 4*tan(x/2) + 2)

Simplificación general [src]

            /x\        
         cot|-|        
            \2/        
-----------------------
              2        
  /       /x\\     4/x\
2*|1 + tan|-|| *cos |-|
  \       \2//      \2/

$$\frac{\cot{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}$$

cot(x/2)/(2*(1 + tan(x/2))^2*cos(x/2)^4)

Denominador racional [src]

/       4/x\        2/x\\    /x\
|1 + tan |-| + 2*tan |-||*cot|-|
\        \2/         \2//    \2/
--------------------------------
             2/x\        /x\    
    2 + 2*tan |-| + 4*tan|-|    
              \2/        \2/

$$\frac{\left(\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2}$$

(1 + tan(x/2)^4 + 2*tan(x/2)^2)*cot(x/2)/(2 + 2*tan(x/2)^2 + 4*tan(x/2))

Unión de expresiones racionales [src]

/       2/x\ /       2/x\\\    /x\
|1 + tan |-|*|2 + tan |-|||*cot|-|
\        \2/ \        \2///    \2/
----------------------------------
     /    /       /x\\    /x\\    
   2*|1 + |2 + tan|-||*tan|-||    
     \    \       \2//    \2//

$$\frac{\left(\left(\tan^{2}{\left(\frac{x}{2} \right)} + 2\right) \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{2 \left(\left(\tan{\left(\frac{x}{2} \right)} + 2\right) \tan{\left(\frac{x}{2} \right)} + 1\right)}$$

(1 + tan(x/2)^2*(2 + tan(x/2)^2))*cot(x/2)/(2*(1 + (2 + tan(x/2))*tan(x/2)))

Abrimos la expresión [src]

            /x\                    4/x\    /x\                 2/x\    /x\    
         cot|-|                 tan |-|*cot|-|            2*tan |-|*cot|-|    
            \2/                     \2/    \2/                  \2/    \2/    
------------------------ + ------------------------ + ------------------------
         2/x\        /x\            2/x\        /x\            2/x\        /x\
2 + 2*tan |-| + 4*tan|-|   2 + 2*tan |-| + 4*tan|-|   2 + 2*tan |-| + 4*tan|-|
          \2/        \2/             \2/        \2/             \2/        \2/

$$\frac{\tan^{4}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2} + \frac{2 \tan^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2} + \frac{\cot{\left(\frac{x}{2} \right)}}{2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2}$$

cot(x/2)/(2 + 2*tan(x/2)^2 + 4*tan(x/2)) + tan(x/2)^4*cot(x/2)/(2 + 2*tan(x/2)^2 + 4*tan(x/2)) + 2*tan(x/2)^2*cot(x/2)/(2 + 2*tan(x/2)^2 + 4*tan(x/2))

Compilar la expresión [src]

/       4/x\        2/x\\    /x\
|1 + tan |-| + 2*tan |-||*cot|-|
\        \2/         \2//    \2/
--------------------------------
             2/x\        /x\    
    2 + 2*tan |-| + 4*tan|-|    
              \2/        \2/

(1 + tan(x/2)^4 + 2*tan(x/2)^2)*cot(x/2)/(2 + 2*tan(x/2)^2 + 4*tan(x/2))

Respuesta numérica [src]

(1.0 + tan(x/2)^4 + 2.0*tan(x/2)^2)*cot(x/2)/(2.0 + 2.0*tan(x/2)^2 + 4.0*tan(x/2))

(1.0 + tan(x/2)^4 + 2.0*tan(x/2)^2)*cot(x/2)/(2.0 + 2.0*tan(x/2)^2 + 4.0*tan(x/2))

Combinatoria [src]

             2       
/       2/x\\     /x\
|1 + tan |-|| *cot|-|
\        \2//     \2/
---------------------
                 2   
     /       /x\\    
   2*|1 + tan|-||    
     \       \2//

$$\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

(1 + tan(x/2)^2)^2*cot(x/2)/(2*(1 + tan(x/2))^2)

Potencias [src]

/       4/x\        2/x\\    /x\
|1 + tan |-| + 2*tan |-||*cot|-|
\        \2/         \2//    \2/
--------------------------------
             2/x\        /x\    
    2 + 2*tan |-| + 4*tan|-|    
              \2/        \2/

/                     4                      2\       
|    /   I*x    -I*x \      /   I*x    -I*x \ |       
|    |   ---    -----|      |   ---    -----| |       
|    |    2       2  |      |    2       2  | |       
|    \- e    + e     /    2*\- e    + e     / |    /x\
|1 + ------------------ - --------------------|*cot|-|
|                    4                     2  |    \2/
|     / I*x    -I*x \       / I*x    -I*x \   |       
|     | ---    -----|       | ---    -----|   |       
|     |  2       2  |       |  2       2  |   |       
\     \e    + e     /       \e    + e     /   /       
------------------------------------------------------
                          2                           
         /   I*x    -I*x \        /   I*x    -I*x \   
         |   ---    -----|        |   ---    -----|   
         |    2       2  |        |    2       2  |   
       2*\- e    + e     /    4*I*\- e    + e     /   
   2 - -------------------- + ---------------------   
                        2          I*x    -I*x        
         / I*x    -I*x \           ---    -----       
         | ---    -----|            2       2         
         |  2       2  |          e    + e            
         \e    + e     /

$$\frac{\left(\frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{4}} - \frac{2 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + 1\right) \cot{\left(\frac{x}{2} \right)}}{- \frac{2 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + \frac{4 i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}} + 2}$$

(1 + (-exp(i*x/2) + exp(-i*x/2))^4/(exp(i*x/2) + exp(-i*x/2))^4 - 2*(-exp(i*x/2) + exp(-i*x/2))^2/(exp(i*x/2) + exp(-i*x/2))^2)*cot(x/2)/(2 - 2*(-exp(i*x/2) + exp(-i*x/2))^2/(exp(i*x/2) + exp(-i*x/2))^2 + 4*i*(-exp(i*x/2) + exp(-i*x/2))/(exp(i*x/2) + exp(-i*x/2)))

Parte trigonométrica [src]

                  4            
     /       2/x\\     /x\     
     |1 + cot |-|| *cot|-|     
     \        \4//     \2/     
-------------------------------
              2               4
  /      1   \  /        2/x\\ 
2*|1 + ------| *|-1 + cot |-|| 
  |       /x\|  \         \4// 
  |    cot|-||                 
  \       \2//

$$\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot{\left(\frac{x}{2} \right)}}{2 \left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)^{2} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}}$$

            /x\        
         csc|-|        
            \2/        
-----------------------
              2        
  /       /x\\     3/x\
2*|1 + tan|-|| *cos |-|
  \       \2//      \2/

$$\frac{\csc{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{3}{\left(\frac{x}{2} \right)}}$$

   3/pi   x\    /x\ 
csc |-- - -|*csc|-| 
    \2    2/    \2/ 
--------------------
                   2
  /       /pi   x\\ 
  |    csc|-- - -|| 
  |       \2    2/| 
2*|1 + -----------| 
  |          /x\  | 
  |       csc|-|  | 
  \          \2/  /

$$\frac{\csc{\left(\frac{x}{2} \right)} \csc^{3}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{2 \left(1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)^{2}}$$

   /       4/x   pi\        2/x   pi\\          
   |    cos |- - --|   4*cos |- - --||          
   |        \2   2 /         \2   2 /|    /x\   
   |1 + ------------ + --------------|*cos|-|   
   |         4/x\        1 + cos(x)  |    \2/   
   |      cos |-|                    |          
   \          \2/                    /          
------------------------------------------------
/         2/x   pi\        /x   pi\\            
|    4*cos |- - --|   4*cos|- - --||            
|          \2   2 /        \2   2 /|    /x   pi\
|2 + -------------- + -------------|*cos|- - --|
|      1 + cos(x)            /x\   |    \2   2 /
|                         cos|-|   |            
\                            \2/   /

$$\frac{\left(1 + \frac{\cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{4}{\left(\frac{x}{2} \right)}} + \frac{4 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} + \frac{4 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$

              /         4/x\         8/x\\     
              |    8*sin |-|   16*sin |-||     
         3    |          \2/          \2/|     
      sin (x)*|1 + --------- + ----------|     
              |        2           4     |     
              \     sin (x)     sin (x)  /     
-----------------------------------------------
                                              2
  /       /    pi\\ /       ___    /    3*pi\\ 
2*|1 - sin|x + --||*|-1 + \/ 2 *sin|x + ----|| 
  \       \    2 // \              \     4  //

$$\frac{\left(\frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\sin^{4}{\left(x \right)}} + \frac{8 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{3}{\left(x \right)}}{2 \left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right) \left(\sqrt{2} \sin{\left(x + \frac{3 \pi}{4} \right)} - 1\right)^{2}}$$

                                        2
                           /       2/x\\ 
                           |1 - cos |-|| 
    1            2         \        \2// 
- ------ + ------------- + --------------
     /x\      /x\    /x\      4/x\    /x\
  tan|-|   cos|-|*sin|-|   cos |-|*tan|-|
     \2/      \2/    \2/       \2/    \2/
-----------------------------------------
                           2             
               /       /x\\              
             2*|1 + tan|-||              
               \       \2//

$$\frac{\frac{\left(1 - \cos^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\cos^{4}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}} - \frac{1}{\tan{\left(\frac{x}{2} \right)}} + \frac{2}{\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

   3/x\     3         3/x\          cos(x)
tan |-| + ------ + tan |-|*cos(x) - ------
    \2/      /x\       \2/             /x\
          tan|-|                    tan|-|
             \2/                       \2/
------------------------------------------
              4*(1 + sin(x))

$$\frac{\cos{\left(x \right)} \tan^{3}{\left(\frac{x}{2} \right)} - \frac{\cos{\left(x \right)}}{\tan{\left(\frac{x}{2} \right)}} + \tan^{3}{\left(\frac{x}{2} \right)} + \frac{3}{\tan{\left(\frac{x}{2} \right)}}}{4 \left(\sin{\left(x \right)} + 1\right)}$$

                 /                              2        \       
                 |                 /       2/x\\     4/x\|       
                 |              32*|1 + tan |-|| *tan |-||       
            3/x\ |       4/x\      \        \2//      \4/|       
       4*tan |-|*|1 + tan |-| + -------------------------|       
             \2/ |        \2/                  4         |       
                 |                /       2/x\\     2/x\ |       
                 |                |1 + tan |-|| *tan |-| |       
                 \                \        \4//      \2/ /       
-----------------------------------------------------------------
                                                                2
               /           2/x\\ /       ___ /       2/x   pi\\\ 
             3 |    1 - tan |-|| |     \/ 2 *|1 - tan |- + --||| 
/       2/x\\  |            \2/| |           \        \2   8 //| 
|1 + tan |-|| *|1 - -----------|*|-1 + ------------------------| 
\        \2//  |           2/x\| |                2/x   pi\    | 
               |    1 + tan |-|| |         1 + tan |- + --|    | 
               \            \2// \                 \2   8 /    /

$$\frac{4 \left(\tan^{4}{\left(\frac{x}{2} \right)} + 1 + \frac{32 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{4}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan^{3}{\left(\frac{x}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} - 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$

 /       4/x\        2/x\\       
 |    sec |-|   2*sec |-||       
 |        \2/         \2/|    /x\
 |1 + ------- + ---------|*csc|-|
 |       4/x\       2/x\ |    \2/
 |    csc |-|    csc |-| |       
 \        \2/        \2/ /       
---------------------------------
/         2/x\        /x\\       
|    2*sec |-|   4*sec|-||       
|          \2/        \2/|    /x\
|2 + --------- + --------|*sec|-|
|        2/x\        /x\ |    \2/
|     csc |-|     csc|-| |       
\         \2/        \2/ /

$$\frac{\left(1 + \frac{2 \sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}} + \frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \sec{\left(\frac{x}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} + \frac{2 \sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}}$$

          3/x\        
       sec |-|        
           \2/        
----------------------
              2       
  /       /x\\     /x\
2*|1 + tan|-|| *sin|-|
  \       \2//     \2/

$$\frac{\sec^{3}{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \sin{\left(\frac{x}{2} \right)}}$$

               /                                2/x\           \
               |                          16*cot |-|           |
/        2/x\\ |       1                         \4/           |
|-1 + cot |-||*|1 + ------- + ---------------------------------|
\         \4// |       4/x\                  /            2/x\\|
               |    cot |-|                2 |    -1 + cot |-|||
               |        \2/   /       2/x\\  |             \2/||
               |              |1 + cot |-|| *|1 + ------------||
               |              \        \4//  |           2/x\ ||
               |                             |    1 + cot |-| ||
               \                             \            \2/ //
----------------------------------------------------------------
     /                               2/x\           \           
     |                         16*cot |-|           |           
     |      4                         \4/           |    /x\    
   2*|2 + ------ + ---------------------------------|*cot|-|    
     |       /x\                  /            2/x\\|    \4/    
     |    cot|-|                2 |    -1 + cot |-|||           
     |       \2/   /       2/x\\  |             \2/||           
     |             |1 + cot |-|| *|1 + ------------||           
     |             \        \4//  |           2/x\ ||           
     |                            |    1 + cot |-| ||           
     \                            \            \2/ //

$$\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \left(1 + \frac{1}{\cot^{4}{\left(\frac{x}{2} \right)}} + \frac{16 \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right)}{2 \left(2 + \frac{4}{\cot{\left(\frac{x}{2} \right)}} + \frac{16 \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right) \cot{\left(\frac{x}{4} \right)}}$$

 /       4/x\        2/x\\       
 |    sin |-|   2*sin |-||       
 |        \2/         \2/|    /x\
 |1 + ------- + ---------|*cos|-|
 |       4/x\       2/x\ |    \2/
 |    cos |-|    cos |-| |       
 \        \2/        \2/ /       
---------------------------------
/         2/x\        /x\\       
|    2*sin |-|   4*sin|-||       
|          \2/        \2/|    /x\
|2 + --------- + --------|*sin|-|
|        2/x\        /x\ |    \2/
|     cos |-|     cos|-| |       
\         \2/        \2/ /

$$\frac{\left(\frac{\sin^{4}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(\frac{x}{2} \right)}} + \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} \right)}}{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}} + \frac{4 \sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} + 2\right) \sin{\left(\frac{x}{2} \right)}}$$

/         4/x\                               \            
|      sec |-|                               |            
|          \2/                  4            |    /x   pi\
|1 + ------------ + -------------------------|*sec|- - --|
|       4/x   pi\   /      1   \    2/x   pi\|    \2   2 /
|    sec |- - --|   |1 + ------|*sec |- - --||            
\        \2   2 /   \    sec(x)/     \2   2 //            
----------------------------------------------------------
   /                                       /x\ \          
   |                                  4*sec|-| |          
   |                4                      \2/ |    /x\   
   |2 + ------------------------- + -----------|*sec|-|   
   |    /      1   \    2/x   pi\      /x   pi\|    \2/   
   |    |1 + ------|*sec |- - --|   sec|- - --||          
   \    \    sec(x)/     \2   2 /      \2   2 //

$$\frac{\left(\frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(\frac{4 \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 2 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}}$$

            4/x\        2/x\     
     1 + tan |-| + 2*tan |-|     
             \2/         \2/     
---------------------------------
/         2/x\        /x\\    /x\
|2 + 2*tan |-| + 4*tan|-||*tan|-|
\          \2/        \2//    \2/

$$\frac{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2\right) \tan{\left(\frac{x}{2} \right)}}$$

   /       4/pi   x\        2/pi   x\\          
   |    csc |-- - -|   2*csc |-- - -||          
   |        \2    2/         \2    2/|    /x\   
   |1 + ------------ + --------------|*csc|-|   
   |         4/x\            2/x\    |    \2/   
   |      csc |-|         csc |-|    |          
   \          \2/             \2/    /          
------------------------------------------------
/         2/pi   x\        /pi   x\\            
|    2*csc |-- - -|   4*csc|-- - -||            
|          \2    2/        \2    2/|    /pi   x\
|2 + -------------- + -------------|*csc|-- - -|
|          2/x\              /x\   |    \2    2/
|       csc |-|           csc|-|   |            
\           \2/              \2/   /

$$\frac{\left(1 + \frac{2 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}} + \frac{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} + \frac{2 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$

/            2/x\            8/x\\            
|       4*sin |-|      16*sin |-||            
|             \2/             \2/|    /pi   x\
|1 + --------------- + ----------|*sin|-- + -|
|           /    pi\       4     |    \2    2/
|    1 + sin|x + --|    sin (x)  |            
\           \    2 /             /            
----------------------------------------------
   /            2/x\           2/x\\          
   |       4*sin |-|      8*sin |-||          
   |             \2/            \2/|    /x\   
   |2 + --------------- + ---------|*sin|-|   
   |           /    pi\     sin(x) |    \2/   
   |    1 + sin|x + --|            |          
   \           \    2 /            /

$$\frac{\left(\frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\sin^{4}{\left(x \right)}} + 1 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}\right) \sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(\frac{8 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 2 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}\right) \sin{\left(\frac{x}{2} \right)}}$$

                 4/pi   x\                      
              csc |-- - -|        2             
                  \2    2/   8*csc (x)          
          1 + ------------ + ---------          
                   4/x\          4/x\           
                csc |-|       csc |-|           
                    \2/           \2/           
------------------------------------------------
                                       2        
                    /          ___    \         
  /         1     \ |        \/ 2     |     3   
2*|1 - -----------|*|-1 + ------------| *csc (x)
  |       /pi    \| |        /     pi\|         
  |    csc|-- - x|| |     csc|-x + --||         
  \       \2     // \        \     4 //

$$\frac{1 + \frac{8 \csc^{2}{\left(x \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}} + \frac{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}}}{2 \left(-1 + \frac{\sqrt{2}}{\csc{\left(- x + \frac{\pi}{4} \right)}}\right)^{2} \left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{3}{\left(x \right)}}$$

                      1                      
---------------------------------------------
               4             2               
  /       2/x\\  /       /x\\     8/x\    /x\
2*|1 - tan |-|| *|1 + tan|-|| *cos |-|*tan|-|
  \        \4//  \       \2//      \4/    \2/

$$\frac{1}{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}$$

                       4                        
------------------------------------------------
                 2                              
/          /x\  \                               
|       sec|-|  |                               
|          \2/  |  /      1              2     \
|1 + -----------| *|------------- + -----------|
|       /x   pi\|  |   /      pi\      /    pi\|
|    sec|- - --||  |sec|2*x - --|   sec|x - --||
\       \2   2 //  \   \      2 /      \    2 //

$$\frac{4}{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \left(\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)}$$

                    sin(x)                   
---------------------------------------------
                                    2        
               /      1        1   \     4/x\
2*(1 - cos(x))*|1 + ------ - ------| *cos |-|
               \    sin(x)   tan(x)/      \2/

$$\frac{\sin{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right) \left(1 - \frac{1}{\tan{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}$$

                  4                   
--------------------------------------
                 2                    
/       /pi   x\\                     
|    csc|-- - -||                     
|       \2    2/|  /   1         2   \
|1 + -----------| *|-------- + ------|
|          /x\  |  \csc(2*x)   csc(x)/
|       csc|-|  |                     
\          \2/  /

$$\frac{4}{\left(1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{1}{\csc{\left(2 x \right)}} + \frac{2}{\csc{\left(x \right)}}\right)}$$

                   1                    
----------------------------------------
                   2                    
  /       /x   pi\\                     
  |    cos|- - --||                     
  |       \2   2 /|     3/x\    /x   pi\
2*|1 + -----------| *cos |-|*cos|- - --|
  |          /x\  |      \2/    \2   2 /
  |       cos|-|  |                     
  \          \2/  /

$$\frac{1}{2 \left(1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$

   3    /       4/x\        2       4/x\\
sin (x)*|1 + tan |-| + 8*csc (x)*sin |-||
        \        \2/                 \2//
-----------------------------------------
                                        2
                /       ___    /    pi\\ 
 2*(1 - cos(x))*|-1 + \/ 2 *cos|x + --|| 
                \              \    4 //

$$\frac{\left(8 \sin^{4}{\left(\frac{x}{2} \right)} \csc^{2}{\left(x \right)} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right) \sin^{3}{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right) \left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2}}$$

   /       4/x   pi\        2/x   pi\\          
   |    cos |- - --|   2*cos |- - --||          
   |        \2   2 /         \2   2 /|    /x\   
   |1 + ------------ + --------------|*cos|-|   
   |         4/x\            2/x\    |    \2/   
   |      cos |-|         cos |-|    |          
   \          \2/             \2/    /          
------------------------------------------------
/         2/x   pi\        /x   pi\\            
|    2*cos |- - --|   4*cos|- - --||            
|          \2   2 /        \2   2 /|    /x   pi\
|2 + -------------- + -------------|*cos|- - --|
|          2/x\              /x\   |    \2   2 /
|       cos |-|           cos|-|   |            
\           \2/              \2/   /

$$\frac{\left(1 + \frac{2 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}} + \frac{\cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{4}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} + \frac{2 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$

            /x\        
         cot|-|        
            \2/        
-----------------------
              2        
  /       /x\\     4/x\
2*|1 + tan|-|| *cos |-|
  \       \2//      \2/

$$\frac{\cot{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}$$

   3/x\    /x   pi\ 
sec |-|*sec|- - --| 
    \2/    \2   2 / 
--------------------
                   2
  /          /x\  \ 
  |       sec|-|  | 
  |          \2/  | 
2*|1 + -----------| 
  |       /x   pi\| 
  |    sec|- - --|| 
  \       \2   2 //

$$\frac{\sec^{3}{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}$$

/                   2/x\ \        
|              4*sin |-| |        
|       4/x\         \2/ |    /x\ 
|1 + tan |-| + ----------|*cos|-| 
\        \2/   1 + cos(x)/    \2/ 
----------------------------------
/                    2/x\ \       
|               4*sin |-| |       
|         /x\         \2/ |    /x\
|2 + 4*tan|-| + ----------|*sin|-|
\         \2/   1 + cos(x)/    \2/

$$\frac{\left(\tan^{4}{\left(\frac{x}{2} \right)} + 1 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(\frac{x}{2} \right)}}{\left(4 \tan{\left(\frac{x}{2} \right)} + 2 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \sin{\left(\frac{x}{2} \right)}}$$

                    4                    
-----------------------------------------
              /                     /x\ \
            2 |                4*tan|-| |
/       /x\\  |  2*tan(x)           \2/ |
|1 + tan|-|| *|----------- + -----------|
\       \2//  |       2             2/x\|
              |1 + tan (x)   1 + tan |-||
              \                      \2//

$$\frac{4}{\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + \frac{4 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

                    4                    
-----------------------------------------
              /                     /x\ \
            2 |                4*cot|-| |
/      1   \  |  2*cot(x)           \2/ |
|1 + ------| *|----------- + -----------|
|       /x\|  |       2             2/x\|
|    cot|-||  |1 + cot (x)   1 + cot |-||
\       \2//  \                      \2//

$$\frac{4}{\left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} + \frac{4 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)}$$

                 4                 
-----------------------------------
            2                      
/       /x\\                       
|1 + tan|-|| *(2*sin(x) + sin(2*x))
\       \2//

$$\frac{4}{\left(2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$

                    /x\                
                 cot|-|                
                    \2/                
---------------------------------------
              2               4        
  /       /x\\  /        2/x\\     8/x\
2*|1 + tan|-|| *|-1 + cot |-|| *sin |-|
  \       \2//  \         \4//      \4/

$$\frac{\cot{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)}}$$

             /       4/x   pi\        4/x   pi\\
             |    cos |- - --|   8*cos |- - --||
   3/    pi\ |        \2   2 /         \2   2 /|
cos |x - --|*|1 + ------------ + --------------|
    \    2 / |         4/x\          2/    pi\ |
             |      cos |-|       cos |x - --| |
             \          \2/           \    2 / /
------------------------------------------------
                                           2    
                   /       ___    /    pi\\     
    2*(1 - cos(x))*|-1 + \/ 2 *cos|x + --||     
                   \              \    4 //

$$\frac{\left(\frac{8 \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1 + \frac{\cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{4}{\left(\frac{x}{2} \right)}}\right) \cos^{3}{\left(x - \frac{\pi}{2} \right)}}{2 \left(1 - \cos{\left(x \right)}\right) \left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2}}$$

              /                               2/x\           \
              |                         16*tan |-|           |
/       2/x\\ |       4/x\                     \4/           |
|1 - tan |-||*|1 + tan |-| + --------------------------------|
\        \4// |        \2/                  /           2/x\\|
              |                           2 |    1 - tan |-|||
              |              /       2/x\\  |            \2/||
              |              |1 + tan |-|| *|1 + -----------||
              |              \        \4//  |           2/x\||
              |                             |    1 + tan |-|||
              \                             \            \2///
--------------------------------------------------------------
    /                                2/x\           \         
    |                          16*tan |-|           |         
    |         /x\                     \4/           |    /x\  
  2*|2 + 4*tan|-| + --------------------------------|*tan|-|  
    |         \2/                  /           2/x\\|    \4/  
    |                            2 |    1 - tan |-|||         
    |               /       2/x\\  |            \2/||         
    |               |1 + tan |-|| *|1 + -----------||         
    |               \        \4//  |           2/x\||         
    |                              |    1 + tan |-|||         
    \                              \            \2///

$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \left(\tan^{4}{\left(\frac{x}{2} \right)} + 1 + \frac{16 \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right)}{2 \left(4 \tan{\left(\frac{x}{2} \right)} + 2 + \frac{16 \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right) \tan{\left(\frac{x}{4} \right)}}$$

           3/x\    6/x\ /       4/x\   2*(1 - cos(x))\     
      4*cot |-|*sin |-|*|1 + tan |-| + --------------|     
            \2/     \2/ \        \2/     1 + cos(x)  /     
-----------------------------------------------------------
                                                          2
             /       ___    2/x   pi\ /        2/x   pi\\\ 
(1 - cos(x))*|-1 + \/ 2 *sin |- + --|*|-1 + cot |- + --||| 
             \               \2   8 / \         \2   8 ///

$$\frac{4 \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right) \sin^{6}{\left(\frac{x}{2} \right)} \cot^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)^{2}}$$

 /         4/x\         8/x\\        
 |    8*sin |-|   16*sin |-||        
 |          \2/          \2/|        
 |1 + --------- + ----------|*sin(x) 
 |        2           4     |        
 \     sin (x)     sin (x)  /        
-------------------------------------
  /         2/x\        4/x\\        
  |    8*sin |-|   8*sin |-||        
  |          \2/         \2/|    2/x\
2*|2 + --------- + ---------|*sin |-|
  |      sin(x)        2    |     \2/
  \                 sin (x) /

$$\frac{\left(\frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\sin^{4}{\left(x \right)}} + \frac{8 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}{2 \left(\frac{8 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \frac{8 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 2\right) \sin^{2}{\left(\frac{x}{2} \right)}}$$

/       1         2   \    /x\
|1 + ------- + -------|*cot|-|
|       4/x\      2/x\|    \2/
|    cot |-|   cot |-||       
\        \2/       \2//       
------------------------------
            2        4        
     2 + ------- + ------     
            2/x\      /x\     
         cot |-|   cot|-|     
             \2/      \2/

$$\frac{\left(1 + \frac{2}{\cot^{2}{\left(\frac{x}{2} \right)}} + \frac{1}{\cot^{4}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}}{2 + \frac{4}{\cot{\left(\frac{x}{2} \right)}} + \frac{2}{\cot^{2}{\left(\frac{x}{2} \right)}}}$$

               2*sin(x)               
--------------------------------------
                                     2
             /       ___    /    pi\\ 
(1 + cos(x))*|-1 + \/ 2 *cos|x + --|| 
             \              \    4 //

$$\frac{2 \sin{\left(x \right)}}{\left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cos{\left(x \right)} + 1\right)}$$

   /       4/pi   x\                            \          
   |    csc |-- - -|                            |          
   |        \2    2/               4            |    /x\   
   |1 + ------------ + -------------------------|*csc|-|   
   |         4/x\      /         1     \    2/x\|    \2/   
   |      csc |-|      |1 + -----------|*csc |-||          
   |          \2/      |       /pi    \|     \2/|          
   |                   |    csc|-- - x||        |          
   \                   \       \2     //        /          
-----------------------------------------------------------
/                                     /pi   x\\            
|                                4*csc|-- - -||            
|                4                    \2    2/|    /pi   x\
|2 + ------------------------- + -------------|*csc|-- - -|
|    /         1     \    2/x\          /x\   |    \2    2/
|    |1 + -----------|*csc |-|       csc|-|   |            
|    |       /pi    \|     \2/          \2/   |            
|    |    csc|-- - x||                        |            
\    \       \2     //                        /

$$\frac{\left(1 + \frac{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}} + \frac{4}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} + \frac{4}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$

/         4/x\                               \            
|      sec |-|                               |            
|          \2/                  4            |    /pi   x\
|1 + ------------ + -------------------------|*sec|-- - -|
|       4/x   pi\   /      1   \    2/x   pi\|    \2    2/
|    sec |- - --|   |1 + ------|*sec |- - --||            
\        \2   2 /   \    sec(x)/     \2   2 //            
----------------------------------------------------------
   /                                       /x\ \          
   |                                  4*sec|-| |          
   |                4                      \2/ |    /x\   
   |2 + ------------------------- + -----------|*sec|-|   
   |    /      1   \    2/x   pi\      /x   pi\|    \2/   
   |    |1 + ------|*sec |- - --|   sec|- - --||          
   \    \    sec(x)/     \2   2 /      \2   2 //

$$\frac{\left(\frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \sec{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(\frac{4 \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 2 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}}$$

                4/x\           2/    pi\       
             sec |-|      8*sec |x - --|       
                 \2/            \    2 /       
       1 + ------------ + --------------       
              4/x   pi\       4/x   pi\        
           sec |- - --|    sec |- - --|        
               \2   2 /        \2   2 /        
-----------------------------------------------
                                 2             
               /          ___   \              
  /      1   \ |        \/ 2    |     3/    pi\
2*|1 - ------|*|-1 + -----------| *sec |x - --|
  \    sec(x)/ |        /    pi\|      \    2 /
               |     sec|x + --||              
               \        \    4 //

$$\frac{\frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 + \frac{8 \sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}}{2 \left(-1 + \frac{\sqrt{2}}{\sec{\left(x + \frac{\pi}{4} \right)}}\right)^{2} \left(1 - \frac{1}{\sec{\left(x \right)}}\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}}$$

                2                
---------------------------------
                         2       
             /       /x\\        
(1 + cos(x))*|1 + tan|-|| *sin(x)
             \       \2//

$$\frac{2}{\left(\cos{\left(x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \sin{\left(x \right)}}$$

/                   2/x\ \              
|              4*sin |-| |              
|       4/x\         \2/ |    /x\    /x\
|1 + tan |-| + ----------|*cos|-|*csc|-|
\        \2/   1 + cos(x)/    \2/    \2/
----------------------------------------
                           2/x\         
                      4*sin |-|         
                /x\         \2/         
       2 + 4*tan|-| + ----------        
                \2/   1 + cos(x)

$$\frac{\left(\tan^{4}{\left(\frac{x}{2} \right)} + 1 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(\frac{x}{2} \right)} \csc{\left(\frac{x}{2} \right)}}{4 \tan{\left(\frac{x}{2} \right)} + 2 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos{\left(x \right)} + 1}}$$

           /                   4/x\\    
           |              8*sin |-||    
      3    |       4/x\         \2/|    
   sin (x)*|1 + tan |-| + ---------|    
           |        \2/       2    |    
           \               sin (x) /    
----------------------------------------
                                       2
               /       ___    /    pi\\ 
2*(1 - cos(x))*|-1 + \/ 2 *cos|x + --|| 
               \              \    4 //

$$\frac{\left(\frac{8 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right) \sin^{3}{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right) \left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2}}$$

/       2/x\\ /       4/x\   2*(1 - cos(x))\
|1 - tan |-||*|1 + tan |-| + --------------|
\        \4// \        \2/     1 + cos(x)  /
--------------------------------------------
    /         /x\   2*(1 - cos(x))\    /x\  
  2*|2 + 4*tan|-| + --------------|*tan|-|  
    \         \2/     1 + cos(x)  /    \4/

$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right)}{2 \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + 4 \tan{\left(\frac{x}{2} \right)} + 2\right) \tan{\left(\frac{x}{4} \right)}}$$

                  4                   
--------------------------------------
               2                      
/         2/x\\                       
|    2*sin |-||                       
|          \2/|                       
|1 + ---------| *(2*sin(x) + sin(2*x))
\      sin(x) /

$$\frac{4}{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right)^{2} \left(2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right)}$$

                        4                         
--------------------------------------------------
                 2                                
/       /x   pi\\                                 
|    cos|- - --||                                 
|       \2   2 /|  /     /    pi\      /      pi\\
|1 + -----------| *|2*cos|x - --| + cos|2*x - --||
|          /x\  |  \     \    2 /      \      2 //
|       cos|-|  |                                 
\          \2/  /

$$\frac{4}{\left(1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right)^{2} \left(2 \cos{\left(x - \frac{\pi}{2} \right)} + \cos{\left(2 x - \frac{\pi}{2} \right)}\right)}$$

/        2/x\\ /       4/x\   2*(1 - cos(x))\    /x\
|-1 + cot |-||*|1 + tan |-| + --------------|*tan|-|
\         \4// \        \2/     1 + cos(x)  /    \4/
----------------------------------------------------
           /         /x\   2*(1 - cos(x))\          
         2*|2 + 4*tan|-| + --------------|          
           \         \2/     1 + cos(x)  /

$$\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{4} \right)}}{2 \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + 4 \tan{\left(\frac{x}{2} \right)} + 2\right)}$$

                  /                              2        \        
                  |                 /       2/x\\     4/x\|        
                  |              32*|1 + cot |-|| *cot |-||        
             3/x\ |       1         \        \2//      \4/|        
        4*cot |-|*|1 + ------- + -------------------------|        
              \2/ |       4/x\                  4         |        
                  |    cot |-|     /       2/x\\     2/x\ |        
                  |        \2/     |1 + cot |-|| *cot |-| |        
                  \                \        \4//      \2/ /        
-------------------------------------------------------------------
                                                                  2
               /            2/x\\ /       ___ /        2/x   pi\\\ 
             3 |    -1 + cot |-|| |     \/ 2 *|-1 + cot |- + --||| 
/       2/x\\  |             \2/| |           \         \2   8 //| 
|1 + cot |-|| *|1 - ------------|*|-1 + -------------------------| 
\        \2//  |           2/x\ | |                 2/x   pi\    | 
               |    1 + cot |-| | |          1 + cot |- + --|    | 
               \            \2/ / \                  \2   8 /    /

$$\frac{4 \left(1 + \frac{1}{\cot^{4}{\left(\frac{x}{2} \right)}} + \frac{32 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cot^{4}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot^{3}{\left(\frac{x}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$

                         4           
            /       2/x\\            
            |1 + tan |-||            
            \        \4//            
-------------------------------------
               4             2       
  /       2/x\\  /       /x\\     /x\
2*|1 - tan |-|| *|1 + tan|-|| *tan|-|
  \        \4//  \       \2//     \2/

$$\frac{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)}}$$

                 sin(x)                
---------------------------------------
                 2                     
  /         2/x\\                      
  |    2*sin |-||                      
  |          \2/|     2/x\    4/pi   x\
4*|1 + ---------| *sin |-|*sin |-- + -|
  \      sin(x) /      \2/     \2    2/

$$\frac{\sin{\left(x \right)}}{4 \left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right)^{2} \sin^{2}{\left(\frac{x}{2} \right)} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}$$

/         4/x\            2/x\  \            
|      sec |-|       2*sec |-|  |            
|          \2/             \2/  |    /x   pi\
|1 + ------------ + ------------|*sec|- - --|
|       4/x   pi\      2/x   pi\|    \2   2 /
|    sec |- - --|   sec |- - --||            
\        \2   2 /       \2   2 //            
---------------------------------------------
   /          2/x\            /x\ \          
   |     2*sec |-|       4*sec|-| |          
   |           \2/            \2/ |    /x\   
   |2 + ------------ + -----------|*sec|-|   
   |       2/x   pi\      /x   pi\|    \2/   
   |    sec |- - --|   sec|- - --||          
   \        \2   2 /      \2   2 //

$$\frac{\left(\frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{2 \sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(\frac{2 \sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{4 \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 2\right) \sec{\left(\frac{x}{2} \right)}}$$

(1 + sec(x/2)^4/sec(x/2 - pi/2)^4 + 2*sec(x/2)^2/sec(x/2 - pi/2)^2)*sec(x/2 - pi/2)/((2 + 2*sec(x/2)^2/sec(x/2 - pi/2)^2 + 4*sec(x/2)/sec(x/2 - pi/2))*sec(x/2))

Denominador común [src]

     /x\      2/x\    /x\        /x\        /x\    /x\                
5*cot|-|   tan |-|*cot|-|   2*cot|-| + 4*cot|-|*tan|-|                
     \2/       \2/    \2/        \2/        \2/    \2/      /x\    /x\
-------- + -------------- - -------------------------- - cot|-|*tan|-|
   2             2                   2/x\        /x\        \2/    \2/
                              1 + tan |-| + 2*tan|-|                  
                                      \2/        \2/

$$- \frac{4 \tan{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)} + 2 \cot{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1} + \frac{\tan^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{2} - \tan{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)} + \frac{5 \cot{\left(\frac{x}{2} \right)}}{2}$$

5*cot(x/2)/2 + tan(x/2)^2*cot(x/2)/2 - (2*cot(x/2) + 4*cot(x/2)*tan(x/2))/(1 + tan(x/2)^2 + 2*tan(x/2)) - cot(x/2)*tan(x/2)

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

Solución

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