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¿cómo vas a descomponer esta expresión en fracciones?:
(z/c+c/z)/(z^2+c^2)/(5*z^9*c)
2*(-1+4*x^2/(1+x^2))/(1+x^2)^2
((z-2)/(4*(z+2)^2))/(z/(2*z-4)-(z^2+4)/(2*z^2-8)-z/(z^2+2*z))
2*(-1+(-1+x)/(2+x))/(2+x)^2
Factorizar el polinomio:
y^7-2*y^5
-y^6+1
y*x^2+i*y^3/3
y^8-x^8
Descomponer al cuadrado:
y^4+9*y^2-3
-y^4-9*y^2+9
-y^4+9*y^2+5
-y^4+9*y^2-1
Expresiones idénticas
cos(x/ cuatro)*sin(x/ cuatro)/(dos *(uno +cos(x/ cuatro)^ dos))+sin(x/ cuatro)^ tres *cos(x/ cuatro)/(dos *(uno +cos(x/ cuatro)^ dos)^ dos)
coseno de (x dividir por 4) multiplicar por seno de (x dividir por 4) dividir por (2 multiplicar por (1 más coseno de (x dividir por 4) al cuadrado )) más seno de (x dividir por 4) al cubo multiplicar por coseno de (x dividir por 4) dividir por (2 multiplicar por (1 más coseno de (x dividir por 4) al cuadrado ) al cuadrado )
coseno de (x dividir por cuatro) multiplicar por seno de (x dividir por cuatro) dividir por (dos multiplicar por (uno más coseno de (x dividir por cuatro) en el grado dos)) más seno de (x dividir por cuatro) en el grado tres multiplicar por coseno de (x dividir por cuatro) dividir por (dos multiplicar por (uno más coseno de (x dividir por cuatro) en el grado dos) en el grado dos)
cos(x/4)*sin(x/4)/(2*(1+cos(x/4)2))+sin(x/4)3*cos(x/4)/(2*(1+cos(x/4)2)2)
cosx/4*sinx/4/2*1+cosx/42+sinx/43*cosx/4/2*1+cosx/422
cos(x/4)*sin(x/4)/(2*(1+cos(x/4)²))+sin(x/4)³*cos(x/4)/(2*(1+cos(x/4)²)²)
cos(x/4)*sin(x/4)/(2*(1+cos(x/4) en el grado 2))+sin(x/4) en el grado 3*cos(x/4)/(2*(1+cos(x/4) en el grado 2) en el grado 2)
cos(x/4)sin(x/4)/(2(1+cos(x/4)^2))+sin(x/4)^3cos(x/4)/(2(1+cos(x/4)^2)^2)
cos(x/4)sin(x/4)/(2(1+cos(x/4)2))+sin(x/4)3cos(x/4)/(2(1+cos(x/4)2)2)
cosx/4sinx/4/21+cosx/42+sinx/43cosx/4/21+cosx/422
cosx/4sinx/4/21+cosx/4^2+sinx/4^3cosx/4/21+cosx/4^2^2
cos(x dividir por 4)*sin(x dividir por 4) dividir por (2*(1+cos(x dividir por 4)^2))+sin(x dividir por 4)^3*cos(x dividir por 4) dividir por (2*(1+cos(x dividir por 4)^2)^2)
Expresiones semejantes
cos(x/4)*sin(x/4)/(2*(1+cos(x/4)^2))+sin(x/4)^3*cos(x/4)/(2*(1-cos(x/4)^2)^2)
cos(x/4)*sin(x/4)/(2*(1+cos(x/4)^2))-sin(x/4)^3*cos(x/4)/(2*(1+cos(x/4)^2)^2)
cos(x/4)*sin(x/4)/(2*(1-cos(x/4)^2))+sin(x/4)^3*cos(x/4)/(2*(1+cos(x/4)^2)^2)
Expresiones con funciones
Coseno cos
cos(2*x)*x/(sin(x)+cos(x))
cos(2*x)/(1-sin(2*x))-(1+tan(x))/(1-tan(x))
cos(x)/atan(x)-sin(x)/((1+x^2)*atan(x)^2)
cos(x)/(2*sqrt(1-sin(x))*sqrt(sin(x)))
cos(x-3/2)/(x^2+1)-2*x*sin(x-3/2)/(x^2+1)^2
Seno sin
sin^2(B)/(1-cos^2(B))+tg^2(B)
sin62+sin^230+sin35+cos^2301/(a*x-1)-1/(a*x+1)
sin(4*x)/(4*cos(4*x))
sin(3*pi/(2-a))-(-1)*cos(a)
sin(a)+3*cos(3*pi/(2+a))
Coseno cos
cos(2*x)*x/(sin(x)+cos(x))
cos(2*x)/(1-sin(2*x))-(1+tan(x))/(1-tan(x))
cos(x)/atan(x)-sin(x)/((1+x^2)*atan(x)^2)
cos(x)/(2*sqrt(1-sin(x))*sqrt(sin(x)))
cos(x-3/2)/(x^2+1)-2*x*sin(x-3/2)/(x^2+1)^2
Seno sin
sin^2(B)/(1-cos^2(B))+tg^2(B)
sin62+sin^230+sin35+cos^2301/(a*x-1)-1/(a*x+1)
sin(4*x)/(4*cos(4*x))
sin(3*pi/(2-a))-(-1)*cos(a)
sin(a)+3*cos(3*pi/(2+a))
Coseno cos
cos(2*x)*x/(sin(x)+cos(x))
cos(2*x)/(1-sin(2*x))-(1+tan(x))/(1-tan(x))
cos(x)/atan(x)-sin(x)/((1+x^2)*atan(x)^2)
cos(x)/(2*sqrt(1-sin(x))*sqrt(sin(x)))
cos(x-3/2)/(x^2+1)-2*x*sin(x-3/2)/(x^2+1)^2
Coseno cos
cos(2*x)*x/(sin(x)+cos(x))
cos(2*x)/(1-sin(2*x))-(1+tan(x))/(1-tan(x))
cos(x)/atan(x)-sin(x)/((1+x^2)*atan(x)^2)
cos(x)/(2*sqrt(1-sin(x))*sqrt(sin(x)))
cos(x-3/2)/(x^2+1)-2*x*sin(x-3/2)/(x^2+1)^2

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

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

Solución

Ha introducido [src]

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

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

(cos(x/4)*sin(x/4))/((2*(1 + cos(x/4)^2))) + (sin(x/4)^3*cos(x/4))/((2*(1 + cos(x/4)^2)^2))

Simplificación general [src]

        /x\  
   2*sin|-|  
        \2/  
-------------
            2
/       /x\\ 
|3 + cos|-|| 
\       \2//

\frac{2 \sin{\left(\frac{x}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

2*sin(x/2)/(3 + cos(x/2))^2

Unión de expresiones racionales [src]

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

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

(1 + cos(x/4)^2 + sin(x/4)^2)*cos(x/4)*sin(x/4)/(2*(1 + cos(x/4)^2)^2)

Respuesta numérica [src]

cos(x/4)*sin(x/4)/(2.0 + 2.0*cos(x/4)^2) + 0.5*sin(x/4)^3*cos(x/4)/(1.0 + cos(x/4)^2)^2

cos(x/4)*sin(x/4)/(2.0 + 2.0*cos(x/4)^2) + 0.5*sin(x/4)^3*cos(x/4)/(1.0 + cos(x/4)^2)^2

Combinatoria [src]

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

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

(1 + cos(x/4)^2 + sin(x/4)^2)*cos(x/4)*sin(x/4)/(2*(1 + cos(x/4)^2)^2)

Denominador común [src]

   3/x\    /x\      3/x\    /x\      /x\    /x\
cos |-|*sin|-| + sin |-|*cos|-| + cos|-|*sin|-|
    \4/    \4/       \4/    \4/      \4/    \4/
-----------------------------------------------
                    4/x\        2/x\           
           2 + 2*cos |-| + 4*cos |-|           
                     \4/         \4/

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

(cos(x/4)^3*sin(x/4) + sin(x/4)^3*cos(x/4) + cos(x/4)*sin(x/4))/(2 + 2*cos(x/4)^4 + 4*cos(x/4)^2)

Denominador racional [src]

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

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

(sin(x/4)^3*(2 + 2*cos(x/4)^2)*cos(x/4) + 2*(1 + cos(x/4)^2)^2*cos(x/4)*sin(x/4))/(2*(1 + cos(x/4)^2)^2*(2 + 2*cos(x/4)^2))

Compilar la expresión [src]

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

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

cos(x/4)*sin(x/4)/(2 + 2*cos(x/4)^2) + sin(x/4)^3*cos(x/4)/(2*(1 + cos(x/4)^2)^2)

Abrimos la expresión [src]

   /x\    /x\            3/x\    /x\     
cos|-|*sin|-|         sin |-|*cos|-|     
   \4/    \4/             \4/    \4/     
------------- + -------------------------
         2/x\            4/x\        2/x\
2 + 2*cos |-|   2 + 2*cos |-| + 4*cos |-|
          \4/             \4/         \4/

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

cos(x/4)*sin(x/4)/(2 + 2*cos(x/4)^2) + sin(x/4)^3*cos(x/4)/(2 + 2*cos(x/4)^4 + 4*cos(x/4)^2)

Potencias [src]

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

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

    / -I*x     I*x\                                        3 / -I*x     I*x\
    | -----    ---| /   -I*x     I*x\     /   -I*x     I*x\  | -----    ---|
    |   4       4 | |   -----    ---|     |   -----    ---|  |   4       4 |
    |e        e   | |     4       4 |     |     4       4 |  |e        e   |
  I*|------ + ----|*\- e      + e   /   I*\- e      + e   / *|------ + ----|
    \  2       2  /                                          \  2       2  /
- ----------------------------------- + ------------------------------------
         /                     2\                                     2     
         |      / -I*x     I*x\ |               /                   2\      
         |      | -----    ---| |               |    / -I*x     I*x\ |      
         |      |   4       4 | |               |    | -----    ---| |      
         |      |e        e   | |               |    |   4       4 | |      
       2*|2 + 2*|------ + ----| |               |    |e        e   | |      
         \      \  2       2  / /            16*|1 + |------ + ----| |      
                                                \    \  2       2  / /

- \frac{i \left(\frac{e^{\frac{i x}{4}}}{2} + \frac{e^{- \frac{i x}{4}}}{2}\right) \left(e^{\frac{i x}{4}} - e^{- \frac{i x}{4}}\right)}{2 \left(2 \left(\frac{e^{\frac{i x}{4}}}{2} + \frac{e^{- \frac{i x}{4}}}{2}\right)^{2} + 2\right)} + \frac{i \left(\frac{e^{\frac{i x}{4}}}{2} + \frac{e^{- \frac{i x}{4}}}{2}\right) \left(e^{\frac{i x}{4}} - e^{- \frac{i x}{4}}\right)^{3}}{16 \left(\left(\frac{e^{\frac{i x}{4}}}{2} + \frac{e^{- \frac{i x}{4}}}{2}\right)^{2} + 1\right)^{2}}

-i*(exp(-i*x/4)/2 + exp(i*x/4)/2)*(-exp(-i*x/4) + exp(i*x/4))/(2*(2 + 2*(exp(-i*x/4)/2 + exp(i*x/4)/2)^2)) + i*(-exp(-i*x/4) + exp(i*x/4))^3*(exp(-i*x/4)/2 + exp(i*x/4)/2)/(16*(1 + (exp(-i*x/4)/2 + exp(i*x/4)/2)^2)^2)

Parte trigonométrica [src]

     4/x\ /       2/x\\    /x\         8/x\    3/x\ /       2/x\\
2*cos |-|*|1 - tan |-||*tan|-|   16*cos |-|*tan |-|*|1 - tan |-||
      \8/ \        \8//    \8/          \8/     \8/ \        \8//
------------------------------ + --------------------------------
                 /x\                                  2          
          3 + cos|-|                      /       /x\\           
                 \2/                      |3 + cos|-||           
                                          \       \2//

\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \cos^{4}{\left(\frac{x}{8} \right)} \tan{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \cos^{8}{\left(\frac{x}{8} \right)} \tan^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

     /x\          
6*sin|-| + sin(x) 
     \2/          
------------------
                 3
/       /pi   x\\ 
|3 + sin|-- + -|| 
\       \2    2//

\frac{6 \sin{\left(\frac{x}{2} \right)} + \sin{\left(x \right)}}{\left(\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 3\right)^{3}}

           /       2/pi   x\\                       /       2/pi   x\\         
           |    csc |-- - -||                       |    csc |-- - -||         
           |        \2    8/|                       |        \2    8/|         
         2*|1 - ------------|                    16*|1 - ------------|         
           |         2/x\   |                       |         2/x\   |         
           |      csc |-|   |                       |      csc |-|   |         
           \          \8/   /                       \          \8/   /         
------------------------------------- + ---------------------------------------
/         1     \    /x\    3/pi   x\                    2                     
|3 + -----------|*csc|-|*csc |-- - -|   /         1     \     3/x\    5/pi   x\
|       /pi   x\|    \8/     \2    8/   |3 + -----------| *csc |-|*csc |-- - -|
|    csc|-- - -||                       |       /pi   x\|      \8/     \2    8/
\       \2    2//                       |    csc|-- - -||                      
                                        \       \2    2//

\frac{2 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{8} \right)}}\right)}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right) \csc{\left(\frac{x}{8} \right)} \csc^{3}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}} + \frac{16 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{8} \right)}}\right)}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right)^{2} \csc^{3}{\left(\frac{x}{8} \right)} \csc^{5}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}}

      4/x \    8/x \ /        2/x\\    /x\           3/x\    8/x \    16/x \ /        2/x\\
32*cot |--|*sin |--|*|-1 + cot |-||*cot|-|   4096*cot |-|*cot |--|*sin  |--|*|-1 + cot |-||
       \16/     \16/ \         \8//    \8/            \8/     \16/      \16/ \         \8//
------------------------------------------ + ----------------------------------------------
                       /x\                                               2                 
                3 + cos|-|                                   /       /x\\                  
                       \2/                                   |3 + cos|-||                  
                                                             \       \2//

\frac{32 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \sin^{8}{\left(\frac{x}{16} \right)} \cot^{4}{\left(\frac{x}{16} \right)} \cot{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{4096 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \sin^{16}{\left(\frac{x}{16} \right)} \cot^{8}{\left(\frac{x}{16} \right)} \cot^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

       /x\           /x\ 
  2*cot|-|     12*cot|-| 
       \2/           \4/ 
----------- + -----------
       2/x\          2/x\
1 + cot |-|   1 + cot |-|
        \2/           \4/
-------------------------
                     3   
   /            2/x\\    
   |    -1 + cot |-||    
   |             \4/|    
   |3 + ------------|    
   |           2/x\ |    
   |    1 + cot |-| |    
   \            \4/ /

\frac{\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{12 \cot{\left(\frac{x}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{3}}

        /          2/x\    \                  /          2/x\    \       
        |       sec |-|    |                  |       sec |-|    |       
        |           \8/    |                  |           \8/    |       
      2*|1 - --------------|               16*|1 - --------------|       
        |       2/  pi   x\|                  |       2/  pi   x\|       
        |    sec |- -- + -||                  |    sec |- -- + -||       
        \        \  2    8//                  \        \  2    8//       
---------------------------------- + ------------------------------------
/      1   \    3/x\    /  pi   x\               2                       
|3 + ------|*sec |-|*sec|- -- + -|   /      1   \     5/x\    3/  pi   x\
|       /x\|     \8/    \  2    8/   |3 + ------| *sec |-|*sec |- -- + -|
|    sec|-||                         |       /x\|      \8/     \  2    8/
\       \2//                         |    sec|-||                        
                                     \       \2//

\frac{2 \left(- \frac{\sec^{2}{\left(\frac{x}{8} \right)}}{\sec^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} + 1\right)}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right) \sec^{3}{\left(\frac{x}{8} \right)} \sec{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} + \frac{16 \left(- \frac{\sec^{2}{\left(\frac{x}{8} \right)}}{\sec^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} + 1\right)}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{5}{\left(\frac{x}{8} \right)} \sec^{3}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}

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

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

                4                                                  8                                
  /       2/x \\     8/x \ /       2/x\\    /x\      /       2/x \\     16/x \    3/x\ /       2/x\\
2*|1 - tan |--|| *cos |--|*|1 - tan |-||*tan|-|   16*|1 - tan |--|| *cos  |--|*tan |-|*|1 - tan |-||
  \        \16//      \16/ \        \8//    \8/      \        \16//       \16/     \8/ \        \8//
----------------------------------------------- + --------------------------------------------------
                          /x\                                                   2                   
                   3 + cos|-|                                       /       /x\\                    
                          \2/                                       |3 + cos|-||                    
                                                                    \       \2//

\frac{16 \left(1 - \tan^{2}{\left(\frac{x}{16} \right)}\right)^{8} \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \cos^{16}{\left(\frac{x}{16} \right)} \tan^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{16} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \cos^{8}{\left(\frac{x}{16} \right)} \tan{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3}

             4/x \ /        1   \                         8/x \ /        1   \       
       32*tan |--|*|-1 + -------|                 4096*tan |--|*|-1 + -------|       
              \16/ |        2/x\|                          \16/ |        2/x\|       
                   |     tan |-||                               |     tan |-||       
                   \         \8//                               \         \8//       
---------------------------------------- + ------------------------------------------
                /           2/x\\                                           2        
              4 |    1 - tan |-||                          /           2/x\\         
/       2/x \\  |            \4/|    /x\                 8 |    1 - tan |-||         
|1 + tan |--|| *|3 + -----------|*tan|-|   /       2/x \\  |            \4/|     3/x\
\        \16//  |           2/x\|    \8/   |1 + tan |--|| *|3 + -----------| *tan |-|
                |    1 + tan |-||          \        \16//  |           2/x\|      \8/
                \            \4//                          |    1 + tan |-||         
                                                           \            \4//

\frac{32 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \tan^{4}{\left(\frac{x}{16} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right) \left(\tan^{2}{\left(\frac{x}{16} \right)} + 1\right)^{4} \tan{\left(\frac{x}{8} \right)}} + \frac{4096 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \tan^{8}{\left(\frac{x}{16} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{2} \left(\tan^{2}{\left(\frac{x}{16} \right)} + 1\right)^{8} \tan^{3}{\left(\frac{x}{8} \right)}}

   /x\    /pi   x\       3/x\    /pi   x\ 
sin|-|*sin|-- + -|    sin |-|*sin|-- + -| 
   \4/    \2    4/        \4/    \2    4/ 
------------------ + ---------------------
         2/pi   x\                       2
2 + 2*sin |-- + -|     /       2/pi   x\\ 
          \2    4/   2*|1 + sin |-- + -|| 
                       \        \2    4//

\frac{\sin{\left(\frac{x}{4} \right)} \sin{\left(\frac{x}{4} + \frac{\pi}{2} \right)}}{2 \sin^{2}{\left(\frac{x}{4} + \frac{\pi}{2} \right)} + 2} + \frac{\sin^{3}{\left(\frac{x}{4} \right)} \sin{\left(\frac{x}{4} + \frac{\pi}{2} \right)}}{2 \left(\sin^{2}{\left(\frac{x}{4} + \frac{\pi}{2} \right)} + 1\right)^{2}}

      4/x \ /        2/x\\    /x\            3/x\    8/x \ /        2/x\\
32*cot |--|*|-1 + cot |-||*cot|-|    4096*cot |-|*cot |--|*|-1 + cot |-||
       \16/ \         \8//    \8/             \8/     \16/ \         \8//
---------------------------------- + ------------------------------------
                /            2/x\\                                     2 
              4 |    -1 + cot |-||                   /            2/x\\  
/       2/x \\  |             \4/|                 8 |    -1 + cot |-||  
|1 + cot |--|| *|3 + ------------|   /       2/x \\  |             \4/|  
\        \16//  |           2/x\ |   |1 + cot |--|| *|3 + ------------|  
                |    1 + cot |-| |   \        \16//  |           2/x\ |  
                \            \4/ /                   |    1 + cot |-| |  
                                                     \            \4/ /

\frac{32 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \cot^{4}{\left(\frac{x}{16} \right)} \cot{\left(\frac{x}{8} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} + 1\right)^{4}} + \frac{4096 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \cot^{8}{\left(\frac{x}{16} \right)} \cot^{3}{\left(\frac{x}{8} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{2} \left(\cot^{2}{\left(\frac{x}{16} \right)} + 1\right)^{8}}

                 /           2/x\    \                                    /           2/x\    \
                 |        cos |-|    |                                    |        cos |-|    |
     3/  pi   x\ |            \8/    |    /x\         3/x\    5/  pi   x\ |            \8/    |
2*cos |- -- + -|*|-1 + --------------|*cos|-|   16*cos |-|*cos |- -- + -|*|-1 + --------------|
      \  2    8/ |        2/  pi   x\|    \8/          \8/     \  2    8/ |        2/  pi   x\|
                 |     cos |- -- + -||                                    |     cos |- -- + -||
                 \         \  2    8//                                    \         \  2    8//
--------------------------------------------- + -----------------------------------------------
                         /x\                                                 2                 
                  3 + cos|-|                                     /       /x\\                  
                         \2/                                     |3 + cos|-||                  
                                                                 \       \2//

\frac{2 \left(\frac{\cos^{2}{\left(\frac{x}{8} \right)}}{\cos^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} - 1\right) \cos{\left(\frac{x}{8} \right)} \cos^{3}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(\frac{\cos^{2}{\left(\frac{x}{8} \right)}}{\cos^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} - 1\right) \cos^{3}{\left(\frac{x}{8} \right)} \cos^{5}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

                1                                      1                   
---------------------------------- + --------------------------------------
/       2   \    /x\    /  pi   x\                  2                      
|2 + -------|*sec|-|*sec|- -- + -|     /       1   \     /x\    3/  pi   x\
|       2/x\|    \4/    \  2    4/   2*|1 + -------| *sec|-|*sec |- -- + -|
|    sec |-||                          |       2/x\|     \4/     \  2    4/
\        \4//                          |    sec |-||                       
                                       \        \4//

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

     3/x\ /       2/x\\    /x\         5/x\    3/x\ /       2/x\\
2*cos |-|*|1 - tan |-||*sin|-|   16*cos |-|*sin |-|*|1 - tan |-||
      \8/ \        \8//    \8/          \8/     \8/ \        \8//
------------------------------ + --------------------------------
                 /x\                                  2          
          3 + cos|-|                      /       /x\\           
                 \2/                      |3 + cos|-||           
                                          \       \2//

\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \sin{\left(\frac{x}{8} \right)} \cos^{3}{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \sin^{3}{\left(\frac{x}{8} \right)} \cos^{5}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

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

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

     3/x\ /        1   \    /x\         3/x\    5/x\ /        1   \
2*sin |-|*|-1 + -------|*cos|-|   16*cos |-|*sin |-|*|-1 + -------|
      \8/ |        2/x\|    \8/          \8/     \8/ |        2/x\|
          |     tan |-||                             |     tan |-||
          \         \8//                             \         \8//
------------------------------- + ---------------------------------
                  /x\                                   2          
           3 + cos|-|                       /       /x\\           
                  \2/                       |3 + cos|-||           
                                            \       \2//

\frac{2 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \sin^{3}{\left(\frac{x}{8} \right)} \cos{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \sin^{5}{\left(\frac{x}{8} \right)} \cos^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

     1             6     
----------- + -----------
   /    pi\      /x   pi\
sec|x - --|   sec|- - --|
   \    2 /      \2   2 /
-------------------------
                  3      
      /      1   \       
      |3 + ------|       
      |       /x\|       
      |    sec|-||       
      \       \2//

\frac{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{6}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right)^{3}}

        /         2/x\ \                            /         2/x\ \
        |      sin |-| |                            |      sin |-| |
   2/x\ |          \4/ |    /x\        3/x\    2/x\ |          \4/ |
sin |-|*|-1 + ---------|*sin|-|   2*sin |-|*sin |-|*|-1 + ---------|
    \8/ |          4/x\|    \4/         \4/     \8/ |          4/x\|
        |     4*sin |-||                            |     4*sin |-||
        \           \8//                            \           \8//
------------------------------- + ----------------------------------
               /pi   x\                                    2        
        3 + sin|-- + -|                   /       /pi   x\\         
               \2    2/                   |3 + sin|-- + -||         
                                          \       \2    2//

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

          /          2/x\   \                       /          2/x\   \        
          |       csc |-|   |                       |       csc |-|   |        
          |           \8/   |                       |           \8/   |        
        2*|-1 + ------------|                    16*|-1 + ------------|        
          |        2/pi   x\|                       |        2/pi   x\|        
          |     csc |-- - -||                       |     csc |-- - -||        
          \         \2    8//                       \         \2    8//        
------------------------------------- + ---------------------------------------
/         1     \    3/x\    /pi   x\                    2                     
|3 + -----------|*csc |-|*csc|-- - -|   /         1     \     5/x\    3/pi   x\
|       /pi   x\|     \8/    \2    8/   |3 + -----------| *csc |-|*csc |-- - -|
|    csc|-- - -||                       |       /pi   x\|      \8/     \2    8/
\       \2    2//                       |    csc|-- - -||                      
                                        \       \2    2//

\frac{2 \left(\frac{\csc^{2}{\left(\frac{x}{8} \right)}}{\csc^{2}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}} - 1\right)}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right) \csc^{3}{\left(\frac{x}{8} \right)} \csc{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}} + \frac{16 \left(\frac{\csc^{2}{\left(\frac{x}{8} \right)}}{\csc^{2}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}} - 1\right)}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right)^{2} \csc^{5}{\left(\frac{x}{8} \right)} \csc^{3}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}}

   /x\    /  pi   x\      3/  pi   x\    /x\
cos|-|*cos|- -- + -|   cos |- -- + -|*cos|-|
   \4/    \  2    4/       \  2    4/    \4/
-------------------- + ---------------------
            2/x\                         2  
   2 + 2*cos |-|            /       2/x\\   
             \4/          2*|1 + cos |-||   
                            \        \4//

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

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

\frac{6 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)} + \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{3}}

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

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

        /x\  
   2*sin|-|  
        \2/  
-------------
            2
/       /x\\ 
|3 + cos|-|| 
\       \2//

\frac{2 \sin{\left(\frac{x}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

                       /         4/x\\                            /         4/x\\
                       |    4*sin |-||                            |    4*sin |-||
     2/x\    4/pi   x\ |          \8/|          6/x\    8/pi   x\ |          \8/|
4*sin |-|*sin |-- + -|*|1 - ---------|   128*sin |-|*sin |-- + -|*|1 - ---------|
      \8/     \2    8/ |        2/x\ |           \8/     \2    8/ |        2/x\ |
                       |     sin |-| |                            |     sin |-| |
                       \         \4/ /                            \         \4/ /
-------------------------------------- + ----------------------------------------
       /       /pi   x\\    /x\                                  2               
       |3 + sin|-- + -||*sin|-|                 /       /pi   x\\     3/x\       
       \       \2    2//    \4/                 |3 + sin|-- + -|| *sin |-|       
                                                \       \2    2//      \4/

\frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{x}{8} \right)}}{\sin^{2}{\left(\frac{x}{4} \right)}} + 1\right) \sin^{2}{\left(\frac{x}{8} \right)} \sin^{4}{\left(\frac{x}{8} + \frac{\pi}{2} \right)}}{\left(\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 3\right) \sin{\left(\frac{x}{4} \right)}} + \frac{128 \left(- \frac{4 \sin^{4}{\left(\frac{x}{8} \right)}}{\sin^{2}{\left(\frac{x}{4} \right)}} + 1\right) \sin^{6}{\left(\frac{x}{8} \right)} \sin^{8}{\left(\frac{x}{8} + \frac{\pi}{2} \right)}}{\left(\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 3\right)^{2} \sin^{3}{\left(\frac{x}{4} \right)}}

          /       2/  pi   x\\                                           /       2/  pi   x\\
          |    cos |- -- + -||                                           |    cos |- -- + -||
     3/x\ |        \  2    8/|    /  pi   x\         5/x\    3/  pi   x\ |        \  2    8/|
2*cos |-|*|1 - --------------|*cos|- -- + -|   16*cos |-|*cos |- -- + -|*|1 - --------------|
      \8/ |          2/x\    |    \  2    8/          \8/     \  2    8/ |          2/x\    |
          |       cos |-|    |                                           |       cos |-|    |
          \           \8/    /                                           \           \8/    /
-------------------------------------------- + ----------------------------------------------
                        /x\                                                2                 
                 3 + cos|-|                                    /       /x\\                  
                        \2/                                    |3 + cos|-||                  
                                                               \       \2//

\frac{2 \left(1 - \frac{\cos^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{8} \right)}}\right) \cos^{3}{\left(\frac{x}{8} \right)} \cos{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(1 - \frac{\cos^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{8} \right)}}\right) \cos^{5}{\left(\frac{x}{8} \right)} \cos^{3}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

     4/x\ /        2/x\\    /x\         3/x\    8/x\ /        2/x\\
2*sin |-|*|-1 + cot |-||*cot|-|   16*cot |-|*sin |-|*|-1 + cot |-||
      \8/ \         \8//    \8/          \8/     \8/ \         \8//
------------------------------- + ---------------------------------
                  /x\                                   2          
           3 + cos|-|                       /       /x\\           
                  \2/                       |3 + cos|-||           
                                            \       \2//

\frac{2 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \sin^{4}{\left(\frac{x}{8} \right)} \cot{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \sin^{8}{\left(\frac{x}{8} \right)} \cot^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

             1                               1               
--------------------------- + -------------------------------
/       2   \    /x\    /x\                  2               
|2 + -------|*csc|-|*sec|-|     /       1   \     3/x\    /x\
|       2/x\|    \4/    \4/   2*|1 + -------| *csc |-|*sec|-|
|    sec |-||                   |       2/x\|      \4/    \4/
\        \4//                   |    sec |-||                
                                \        \4//

\frac{1}{\left(2 + \frac{2}{\sec^{2}{\left(\frac{x}{4} \right)}}\right) \csc{\left(\frac{x}{4} \right)} \sec{\left(\frac{x}{4} \right)}} + \frac{1}{2 \left(1 + \frac{1}{\sec^{2}{\left(\frac{x}{4} \right)}}\right)^{2} \csc^{3}{\left(\frac{x}{4} \right)} \sec{\left(\frac{x}{4} \right)}}

                 4                                                   8                                
  /        2/x \\     8/x \ /       2/x\\    /x\      /        2/x \\     16/x \    3/x\ /       2/x\\
2*|-1 + cot |--|| *sin |--|*|1 - tan |-||*tan|-|   16*|-1 + cot |--|| *sin  |--|*tan |-|*|1 - tan |-||
  \         \16//      \16/ \        \8//    \8/      \         \16//       \16/     \8/ \        \8//
------------------------------------------------ + ---------------------------------------------------
                          /x\                                                     2                   
                   3 + cos|-|                                         /       /x\\                    
                          \2/                                         |3 + cos|-||                    
                                                                      \       \2//

\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{16} \right)} \tan{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{x}{16} \right)} \tan^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}

      8/x \    4/x \ /        1   \           16/x \    8/x \ /        1   \
32*cos |--|*tan |--|*|-1 + -------|   4096*cos  |--|*tan |--|*|-1 + -------|
       \16/     \16/ |        2/x\|             \16/     \16/ |        2/x\|
                     |     tan |-||                           |     tan |-||
                     \         \8//                           \         \8//
----------------------------------- + --------------------------------------
        /       /x\\    /x\                               2                 
        |3 + cos|-||*tan|-|                   /       /x\\     3/x\         
        \       \2//    \8/                   |3 + cos|-|| *tan |-|         
                                              \       \2//      \8/

\frac{32 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \cos^{8}{\left(\frac{x}{16} \right)} \tan^{4}{\left(\frac{x}{16} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right) \tan{\left(\frac{x}{8} \right)}} + \frac{4096 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \cos^{16}{\left(\frac{x}{16} \right)} \tan^{8}{\left(\frac{x}{16} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2} \tan^{3}{\left(\frac{x}{8} \right)}}

       /x\           /x\ 
  2*tan|-|     12*tan|-| 
       \2/           \4/ 
----------- + -----------
       2/x\          2/x\
1 + tan |-|   1 + tan |-|
        \2/           \4/
-------------------------
                     3   
    /           2/x\\    
    |    1 - tan |-||    
    |            \4/|    
    |3 + -----------|    
    |           2/x\|    
    |    1 + tan |-||    
    \            \4//

\frac{\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{12 \tan{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{3}}

                      4                                            8                   
       /        2/x \\  /       1   \               /        2/x \\  /       1   \     
     2*|-1 + cot |--|| *|1 - -------|            16*|-1 + cot |--|| *|1 - -------|     
       \         \16//  |       2/x\|               \         \16//  |       2/x\|     
                        |    cot |-||                                |    cot |-||     
                        \        \8//                                \        \8//     
----------------------------------------- + -------------------------------------------
                /            2/x\\                                            2        
              4 |    -1 + cot |-||                          /            2/x\\         
/       2/x \\  |             \4/|    /x\                 8 |    -1 + cot |-||         
|1 + cot |--|| *|3 + ------------|*cot|-|   /       2/x \\  |             \4/|     3/x\
\        \16//  |           2/x\ |    \8/   |1 + cot |--|| *|3 + ------------| *cot |-|
                |    1 + cot |-| |          \        \16//  |           2/x\ |      \8/
                \            \4/ /                          |    1 + cot |-| |         
                                                            \            \4/ /

\frac{2 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{8} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} - 1\right)^{4}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} + 1\right)^{4} \cot{\left(\frac{x}{8} \right)}} + \frac{16 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{8} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} - 1\right)^{8}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{2} \left(\cot^{2}{\left(\frac{x}{16} \right)} + 1\right)^{8} \cot^{3}{\left(\frac{x}{8} \right)}}

                  1                                         1                    
------------------------------------- + -----------------------------------------
/         2      \    /x\    /pi   x\                       2                    
|2 + ------------|*csc|-|*csc|-- - -|     /         1      \     3/x\    /pi   x\
|       2/pi   x\|    \4/    \2    4/   2*|1 + ------------| *csc |-|*csc|-- - -|
|    csc |-- - -||                        |       2/pi   x\|      \4/    \2    4/
\        \2    4//                        |    csc |-- - -||                     
                                          \        \2    4//

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

     /x\         
6*sin|-| + sin(x)
     \2/         
-----------------
              3  
  /       /x\\   
  |3 + cos|-||   
  \       \2//

\frac{6 \sin{\left(\frac{x}{2} \right)} + \sin{\left(x \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{3}}

       /        2/  pi   x\\                  /        2/  pi   x\\      
       |     sec |- -- + -||                  |     sec |- -- + -||      
       |         \  2    8/|                  |         \  2    8/|      
     2*|-1 + --------------|               16*|-1 + --------------|      
       |           2/x\    |                  |           2/x\    |      
       |        sec |-|    |                  |        sec |-|    |      
       \            \8/    /                  \            \8/    /      
---------------------------------- + ------------------------------------
/      1   \    /x\    3/  pi   x\               2                       
|3 + ------|*sec|-|*sec |- -- + -|   /      1   \     3/x\    5/  pi   x\
|       /x\|    \8/     \  2    8/   |3 + ------| *sec |-|*sec |- -- + -|
|    sec|-||                         |       /x\|      \8/     \  2    8/
\       \2//                         |    sec|-||                        
                                     \       \2//

\frac{2 \left(-1 + \frac{\sec^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{8} \right)}}\right)}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{8} \right)} \sec^{3}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} + \frac{16 \left(-1 + \frac{\sec^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{8} \right)}}\right)}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{3}{\left(\frac{x}{8} \right)} \sec^{5}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}

   1        6     
 ------ + ------  
 csc(x)      /x\  
          csc|-|  
             \2/  
------------------
                 3
/         1     \ 
|3 + -----------| 
|       /pi   x\| 
|    csc|-- - -|| 
\       \2    2//

\frac{\frac{1}{\csc{\left(x \right)}} + \frac{6}{\csc{\left(\frac{x}{2} \right)}}}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right)^{3}}

                4                                         8                      
  /       2/x \\  /       2/x\\    /x\      /       2/x \\     3/x\ /       2/x\\
2*|1 - tan |--|| *|1 - tan |-||*tan|-|   16*|1 - tan |--|| *tan |-|*|1 - tan |-||
  \        \16//  \        \8//    \8/      \        \16//      \8/ \        \8//
-------------------------------------- + ----------------------------------------
                  /           2/x\\                                          2   
                4 |    1 - tan |-||                         /           2/x\\    
  /       2/x \\  |            \4/|                       8 |    1 - tan |-||    
  |1 + tan |--|| *|3 + -----------|         /       2/x \\  |            \4/|    
  \        \16//  |           2/x\|         |1 + tan |--|| *|3 + -----------|    
                  |    1 + tan |-||         \        \16//  |           2/x\|    
                  \            \4//                         |    1 + tan |-||    
                                                            \            \4//

\frac{16 \left(1 - \tan^{2}{\left(\frac{x}{16} \right)}\right)^{8} \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \tan^{3}{\left(\frac{x}{8} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{2} \left(\tan^{2}{\left(\frac{x}{16} \right)} + 1\right)^{8}} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{16} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \tan{\left(\frac{x}{8} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right) \left(\tan^{2}{\left(\frac{x}{16} \right)} + 1\right)^{4}}

2*(1 - tan(x/16)^2)^4*(1 - tan(x/8)^2)*tan(x/8)/((1 + tan(x/16)^2)^4*(3 + (1 - tan(x/4)^2)/(1 + tan(x/4)^2))) + 16*(1 - tan(x/16)^2)^8*tan(x/8)^3*(1 - tan(x/8)^2)/((1 + tan(x/16)^2)^8*(3 + (1 - tan(x/4)^2)/(1 + tan(x/4)^2))^2)

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

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