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((z-2)/(4*z^2+16*z+16))/((z/(2*z-4))-((z^2+4)/(2*z^2-8))-(z/(z^2+2*z)))
(z^2-4*z+16)/(16*z^2-1)*(4*z^2+z)/(z^3+64)-(z+4)/(4*z^2-z)
z/(x+y)+x/(z+y)+y/(z+x)
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Factorizar el polinomio:
x^8+y^8
x^4-x^3
x^4+x^3-x+1
x^4+1
Descomponer al cuadrado:
y^4-y^2+11
y^4-y^2+1
-y^4+9*y^2+6
y^4-3*y^2+1
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Expresiones semejantes
sina/(sin^3a-3cos^3a)
Expresiones con funciones
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sin(pi+a)+cos(pi+a)/cos(pi/2-a)-sin(3pi/2-a)1/(a*x-1)-1/(a*x+1)
sin(2*x)/((19*sin(x)))

Simplificación de expresiones/ Descomposición en fracciones simples/ sina/(sin^3a+3cos^3a)

¿Cómo vas a descomponer esta sina/(sin^3a+3cos^3a) expresión en fracciones?

Solución

Ha introducido [src]

       sin(a)      
-------------------
   3           3   
sin (a) + 3*cos (a)

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

sin(a)/(sin(a)^3 + 3*cos(a)^3)

Respuesta numérica [src]

sin(a)/(sin(a)^3 + 3.0*cos(a)^3)

sin(a)/(sin(a)^3 + 3.0*cos(a)^3)

Potencias [src]

               /   -I*a    I*a\            
            -I*\- e     + e   /            
-------------------------------------------
  /                3                     3\
  |  / I*a    -I*a\      /   -I*a    I*a\ |
  |  |e      e    |    I*\- e     + e   / |
2*|3*|---- + -----|  + -------------------|
  \  \ 2       2  /             8         /

- \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(3 \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{3} + \frac{i \left(e^{i a} - e^{- i a}\right)^{3}}{8}\right)}

-i*(-exp(-i*a) + exp(i*a))/(2*(3*(exp(i*a)/2 + exp(-i*a)/2)^3 + i*(-exp(-i*a) + exp(i*a))^3/8))

Parte trigonométrica [src]

                               2                                
                  /       2/a\\     4/a\    /a\                 
                2*|1 - tan |-|| *cos |-|*tan|-|                 
                  \        \4//      \4/    \2/                 
----------------------------------------------------------------
                                       3              6         
  sin(3*a)   3*sin(a)     /       2/a\\  /       2/a\\     12/a\
- -------- + -------- + 3*|1 - tan |-|| *|1 - tan |-|| *cos  |-|
     4          4         \        \2//  \        \4//       \4/

\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2} \cos^{4}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} \right)}}{3 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{6} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3} \cos^{12}{\left(\frac{a}{4} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}

                                 2                                  
--------------------------------------------------------------------
/                                              3\                   
|                            /       2/pi   a\\ |                   
|                            |    csc |-- - -|| |                   
|                            |        \2    2/| |                   
|                          3*|1 - ------------| |                   
|                            |         2/a\   | |                   
|                            |      csc |-|   | |                   
|      1           3         \          \2/   / |    /a\    /pi   a\
|- ---------- + -------- + ---------------------|*csc|-|*csc|-- - -|
|  4*csc(3*a)   4*csc(a)           6/pi   a\    |    \2/    \2    2/
|                               csc |-- - -|    |                   
\                                   \2    2/    /

\frac{2}{\left(\frac{3 \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{3}}{\csc^{6}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{1}{4 \csc{\left(3 a \right)}} + \frac{3}{4 \csc{\left(a \right)}}\right) \csc{\left(\frac{a}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}

         /    pi\       
      cos|a - --|       
         \    2 /       
------------------------
   3/    pi\        3   
cos |a - --| + 3*cos (a)
    \    2 /

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

                  /a\    /a\              
             2*cos|-|*sin|-|              
                  \2/    \2/              
------------------------------------------
                              3       6/a\
                        24*cos (a)*cos |-|
  sin(3*a)   3*sin(a)                  \2/
- -------- + -------- + ------------------
     4          4                     3   
                          (1 + cos(a))

\frac{2 \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)}}{\frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4} + \frac{24 \cos^{6}{\left(\frac{a}{2} \right)} \cos^{3}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right)^{3}}}

                                                       2                                        
                                         /        2/a\\                                         
                                       2*|-1 + cot |-||                                         
                                         \         \4//                                         
------------------------------------------------------------------------------------------------
               /                                                       3               6\       
               |                                          /       1   \  /        2/a\\ |       
               |                                        3*|1 - -------| *|-1 + cot |-|| |       
               |          /3*a\                /a\        |       2/a\|  \         \4// |       
             2 |       cot|---|           3*cot|-|        |    cot |-||                 |       
/       2/a\\  |          \ 2 /                \2/        \        \2//                 |    /a\
|1 + cot |-|| *|- ----------------- + --------------- + --------------------------------|*cot|-|
\        \4//  |    /       2/3*a\\     /       2/a\\                         6         |    \2/
               |  2*|1 + cot |---||   2*|1 + cot |-||            /       2/a\\          |       
               |    \        \ 2 //     \        \2//            |1 + cot |-||          |       
               \                                                 \        \4//          /

\frac{2 \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\frac{3 \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{6}} - \frac{\cot{\left(\frac{3 a}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right)} + \frac{3 \cot{\left(\frac{a}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}\right) \cot{\left(\frac{a}{2} \right)}}

                  4*sin(a)                  
--------------------------------------------
-sin(3*a) + 3*cos(3*a) + 3*sin(a) + 9*cos(a)

\frac{4 \sin{\left(a \right)}}{3 \sin{\left(a \right)} - \sin{\left(3 a \right)} + 9 \cos{\left(a \right)} + 3 \cos{\left(3 a \right)}}

                           /a\    /a   pi\                     
                      2*cos|-|*cos|- - --|                     
                           \2/    \2   2 /                     
---------------------------------------------------------------
                                                      3        
     /      pi\        /    pi\     /       2/a   pi\\         
  cos|3*a - --|   3*cos|a - --|     |    cos |- - --||         
     \      2 /        \    2 /     |        \2   2 /|     6/a\
- ------------- + ------------- + 3*|1 - ------------| *cos |-|
        4               4           |         2/a\   |      \2/
                                    |      cos |-|   |         
                                    \          \2/   /

\frac{2 \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{3 \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \cos^{6}{\left(\frac{a}{2} \right)} + \frac{3 \cos{\left(a - \frac{\pi}{2} \right)}}{4} - \frac{\cos{\left(3 a - \frac{\pi}{2} \right)}}{4}}

                          /a\                     
                     2*cot|-|                     
                          \2/                     
--------------------------------------------------
              /                3                 \
              |  /        2/a\\           3/a\   |
              |3*|-1 + cot |-||      8*cot |-|   |
/       2/a\\ |  \         \2//            \2/   |
|1 + cot |-||*|----------------- + --------------|
\        \2// |               3                 3|
              |  /       2/a\\     /       2/a\\ |
              |  |1 + cot |-||     |1 + cot |-|| |
              \  \        \2//     \        \2// /

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

                                       2                                       
-------------------------------------------------------------------------------
/                                                         3\                   
|                                      /        2/a   pi\\ |                   
|                                      |     sec |- - --|| |                   
|                                      |         \2   2 /| |                   
|                                    3*|-1 + ------------| |                   
|                                      |          2/a\   | |                   
|                                      |       sec |-|   | |                   
|         1                3           \           \2/   / |    /a\    /a   pi\
|- --------------- + ------------- + ----------------------|*sec|-|*sec|- - --|
|       /      pi\        /    pi\           6/a   pi\     |    \2/    \2   2 /
|  4*sec|3*a - --|   4*sec|a - --|        sec |- - --|     |                   
\       \      2 /        \    2 /            \2   2 /     /

\frac{2}{\left(\frac{3 \left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{3}}{\sec^{6}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - \frac{1}{4 \sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{3}{4 \sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}

                      /a\    /a\                 
                 2*cos|-|*sin|-|                 
                      \2/    \2/                 
-------------------------------------------------
                                        3        
  sin(3*a)   3*sin(a)     /        1   \     6/a\
- -------- + -------- + 3*|-1 + -------| *sin |-|
     4          4         |        2/a\|      \2/
                          |     tan |-||         
                          \         \2//

\frac{2 \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)}}{3 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \sin^{6}{\left(\frac{a}{2} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}

                                      2                                       
------------------------------------------------------------------------------
/                                                        3\                   
|                                      /         2/a\   \ |                   
|                                      |      sec |-|   | |                   
|                                      |          \2/   | |                   
|                                    3*|1 - ------------| |                   
|                                      |       2/a   pi\| |                   
|                                      |    sec |- - --|| |                   
|         1                3           \        \2   2 // |    /a\    /a   pi\
|- --------------- + ------------- + ---------------------|*sec|-|*sec|- - --|
|       /      pi\        /    pi\             6/a\       |    \2/    \2   2 /
|  4*sec|3*a - --|   4*sec|a - --|          sec |-|       |                   
\       \      2 /        \    2 /              \2/       /

\frac{2}{\left(\frac{3 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{3}}{\sec^{6}{\left(\frac{a}{2} \right)}} - \frac{1}{4 \sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{3}{4 \sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}

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

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

                                               2                                        
                                  /       2/a\\     /a\                                 
                                2*|1 - tan |-|| *tan|-|                                 
                                  \        \4//     \2/                                 
----------------------------------------------------------------------------------------
               /                                                       3              6\
               |          /3*a\                /a\        /       2/a\\  /       2/a\\ |
             2 |       tan|---|           3*tan|-|      3*|1 - tan |-|| *|1 - tan |-|| |
/       2/a\\  |          \ 2 /                \2/        \        \2//  \        \4// |
|1 + tan |-|| *|- ----------------- + --------------- + -------------------------------|
\        \4//  |    /       2/3*a\\     /       2/a\\                         6        |
               |  2*|1 + tan |---||   2*|1 + tan |-||            /       2/a\\         |
               |    \        \ 2 //     \        \2//            |1 + tan |-||         |
               \                                                 \        \4//         /

\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2} \tan{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\frac{3 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{6} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{6}} - \frac{\tan{\left(\frac{3 a}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)} + \frac{3 \tan{\left(\frac{a}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}\right)}

                     2/a\    /a\                
                2*sin |-|*cot|-|                
                      \2/    \2/                
------------------------------------------------
                                       3        
-sin(3*a) + 3*sin(a)     /        2/a\\     6/a\
-------------------- + 3*|-1 + cot |-|| *sin |-|
         4               \         \2//      \2/

\frac{2 \sin^{2}{\left(\frac{a}{2} \right)} \cot{\left(\frac{a}{2} \right)}}{\frac{3 \sin{\left(a \right)} - \sin{\left(3 a \right)}}{4} + 3 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{3} \sin^{6}{\left(\frac{a}{2} \right)}}

                                  2                                  
---------------------------------------------------------------------
/                                               3\                   
|                            /          2/a\   \ |                   
|                            |       csc |-|   | |                   
|                            |           \2/   | |                   
|                          3*|-1 + ------------| |                   
|                            |        2/pi   a\| |                   
|                            |     csc |-- - -|| |                   
|      1           3         \         \2    2// |    /a\    /pi   a\
|- ---------- + -------- + ----------------------|*csc|-|*csc|-- - -|
|  4*csc(3*a)   4*csc(a)             6/a\        |    \2/    \2    2/
|                                 csc |-|        |                   
\                                     \2/        /

\frac{2}{\left(\frac{3 \left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{3}}{\csc^{6}{\left(\frac{a}{2} \right)}} - \frac{1}{4 \csc{\left(3 a \right)}} + \frac{3}{4 \csc{\left(a \right)}}\right) \csc{\left(\frac{a}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}

               1               
-------------------------------
/   1           3      \       
|------- + ------------|*csc(a)
|   3         3/pi    \|       
|csc (a)   csc |-- - a||       
\              \2     //

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

                     2/a\    /a\               
                2*cos |-|*tan|-|               
                      \2/    \2/               
-----------------------------------------------
                                      3        
-sin(3*a) + 3*sin(a)     /       2/a\\     6/a\
-------------------- + 3*|1 - tan |-|| *cos |-|
         4               \        \2//      \2/

\frac{2 \cos^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} \right)}}{3 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3} \cos^{6}{\left(\frac{a}{2} \right)} + \frac{3 \sin{\left(a \right)} - \sin{\left(3 a \right)}}{4}}

                  /a\    /a\              
             2*cos|-|*sin|-|              
                  \2/    \2/              
------------------------------------------
                              3       6/a\
                        24*cos (a)*sin |-|
  sin(3*a)   3*sin(a)                  \2/
- -------- + -------- - ------------------
     4          4                      3  
                          (-1 + cos(a))

\frac{2 \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)}}{\frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4} - \frac{24 \sin^{6}{\left(\frac{a}{2} \right)} \cos^{3}{\left(a \right)}}{\left(\cos{\left(a \right)} - 1\right)^{3}}}

                       2/a\    4/a\    /a\                  
                  8*cot |-|*sin |-|*cot|-|                  
                        \4/     \4/    \2/                  
------------------------------------------------------------
                                          3                 
  sin(3*a)   3*sin(a)       /        2/a\\     6/a\    12/a\
- -------- + -------- + 192*|-1 + cot |-|| *cot |-|*sin  |-|
     4          4           \         \2//      \4/      \4/

\frac{8 \sin^{4}{\left(\frac{a}{4} \right)} \cot^{2}{\left(\frac{a}{4} \right)} \cot{\left(\frac{a}{2} \right)}}{192 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{3} \sin^{12}{\left(\frac{a}{4} \right)} \cot^{6}{\left(\frac{a}{4} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}

                               4/a\    2/a\                          
                          8*cos |-|*tan |-|                          
                                \4/     \4/                          
---------------------------------------------------------------------
/                                          3                 \       
|  sin(3*a)   3*sin(a)       /        1   \     12/a\    6/a\|    /a\
|- -------- + -------- + 192*|-1 + -------| *cos  |-|*tan |-||*tan|-|
|     4          4           |        2/a\|       \4/     \4/|    \2/
|                            |     tan |-||                  |       
\                            \         \2//                  /

\frac{8 \cos^{4}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{4} \right)}}{\left(192 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \cos^{12}{\left(\frac{a}{4} \right)} \tan^{6}{\left(\frac{a}{4} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}\right) \tan{\left(\frac{a}{2} \right)}}

                              /a\    /a   pi\                        
                         2*cos|-|*cos|- - --|                        
                              \2/    \2   2 /                        
---------------------------------------------------------------------
                                                       3             
     /      pi\        /    pi\     /          2/a\   \              
  cos|3*a - --|   3*cos|a - --|     |       cos |-|   |              
     \      2 /        \    2 /     |           \2/   |     6/a   pi\
- ------------- + ------------- + 3*|-1 + ------------| *cos |- - --|
        4               4           |        2/a   pi\|      \2   2 /
                                    |     cos |- - --||              
                                    \         \2   2 //

\frac{2 \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{3 \left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{3} \cos^{6}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} + \frac{3 \cos{\left(a - \frac{\pi}{2} \right)}}{4} - \frac{\cos{\left(3 a - \frac{\pi}{2} \right)}}{4}}

                       sin(a)                      
---------------------------------------------------
                                          3        
                          /         2    \         
  sin(3*a)   3*sin(a)     |      sin (a) |     6/a\
- -------- + -------- + 3*|-1 + ---------| *sin |-|
     4          4         |          4/a\|      \2/
                          |     4*sin |-||         
                          \           \2//

\frac{\sin{\left(a \right)}}{3 \left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{3} \sin^{6}{\left(\frac{a}{2} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}

                                       2/a\    /a\                                  
                                  8*cot |-|*cot|-|                                  
                                        \4/    \2/                                  
------------------------------------------------------------------------------------
               /                                                          3        \
               |          /3*a\                /a\          /        2/a\\     6/a\|
             2 |       cot|---|           3*cot|-|      192*|-1 + cot |-|| *cot |-||
/       2/a\\  |          \ 2 /                \2/          \         \2//      \4/|
|1 + cot |-|| *|- ----------------- + --------------- + ---------------------------|
\        \4//  |    /       2/3*a\\     /       2/a\\                       6      |
               |  2*|1 + cot |---||   2*|1 + cot |-||          /       2/a\\       |
               |    \        \ 2 //     \        \2//          |1 + cot |-||       |
               \                                               \        \4//       /

\frac{8 \cot^{2}{\left(\frac{a}{4} \right)} \cot{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(- \frac{\cot{\left(\frac{3 a}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right)} + \frac{3 \cot{\left(\frac{a}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} + \frac{192 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{3} \cot^{6}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{6}}\right)}

                     /a\    /a\                 
                2*cos|-|*sin|-|                 
                     \2/    \2/                 
------------------------------------------------
                                       3        
  sin(3*a)   3*sin(a)     /       2/a\\     6/a\
- -------- + -------- + 3*|1 - tan |-|| *cos |-|
     4          4         \        \2//      \2/

\frac{2 \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)}}{3 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3} \cos^{6}{\left(\frac{a}{2} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}

                 1                  
------------------------------------
/     1            3   \    /    pi\
|------------ + -------|*sec|a - --|
|   3/    pi\      3   |    \    2 /
|sec |a - --|   sec (a)|            
\    \    2 /          /

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

                          /a\                    
                     2*tan|-|                    
                          \2/                    
-------------------------------------------------
              /               3                 \
              |  /       2/a\\           3/a\   |
              |3*|1 - tan |-||      8*tan |-|   |
/       2/a\\ |  \        \2//            \2/   |
|1 + tan |-||*|---------------- + --------------|
\        \2// |              3                 3|
              | /       2/a\\     /       2/a\\ |
              | |1 + tan |-||     |1 + tan |-|| |
              \ \        \2//     \        \2// /

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

                                              2/a\                                         
                                         8*tan |-|                                         
                                               \4/                                         
-------------------------------------------------------------------------------------------
               /                                                          3        \       
               |                                            /        1   \     6/a\|       
               |                                        192*|-1 + -------| *tan |-||       
               |          /3*a\                /a\          |        2/a\|      \4/|       
             2 |       tan|---|           3*tan|-|          |     tan |-||         |       
/       2/a\\  |          \ 2 /                \2/          \         \2//         |    /a\
|1 + tan |-|| *|- ----------------- + --------------- + ---------------------------|*tan|-|
\        \4//  |    /       2/3*a\\     /       2/a\\                       6      |    \2/
               |  2*|1 + tan |---||   2*|1 + tan |-||          /       2/a\\       |       
               |    \        \ 2 //     \        \2//          |1 + tan |-||       |       
               \                                               \        \4//       /

\frac{8 \tan^{2}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\frac{192 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \tan^{6}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{6}} - \frac{\tan{\left(\frac{3 a}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)} + \frac{3 \tan{\left(\frac{a}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}\right) \tan{\left(\frac{a}{2} \right)}}

            1             
--------------------------
/   1         3   \       
|------- + -------|*csc(a)
|   3         3   |       
\csc (a)   sec (a)/

\frac{1}{\left(\frac{3}{\sec^{3}{\left(a \right)}} + \frac{1}{\csc^{3}{\left(a \right)}}\right) \csc{\left(a \right)}}

                          2/a\    2/pi   a\                     
                     4*sin |-|*sin |-- + -|                     
                           \2/     \2    2/                     
----------------------------------------------------------------
/                                         3             \       
|                          /         4/a\\              |       
|                          |    4*sin |-||              |       
|  sin(3*a)   3*sin(a)     |          \2/|     6/pi   a\|       
|- -------- + -------- + 3*|1 - ---------| *sin |-- + -||*sin(a)
|     4          4         |        2    |      \2    2/|       
\                          \     sin (a) /              /

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

                                 2                               
                   /        2/a\\     4/a\    /a\                
                 2*|-1 + cot |-|| *sin |-|*tan|-|                
                   \         \4//      \4/    \2/                
-----------------------------------------------------------------
                                       3               6         
  sin(3*a)   3*sin(a)     /       2/a\\  /        2/a\\     12/a\
- -------- + -------- + 3*|1 - tan |-|| *|-1 + cot |-|| *sin  |-|
     4          4         \        \2//  \         \4//       \4/

\frac{2 \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} \right)}}{3 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3} \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{a}{4} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}

2*(-1 + cot(a/4)^2)^2*sin(a/4)^4*tan(a/2)/(-sin(3*a)/4 + 3*sin(a)/4 + 3*(1 - tan(a/2)^2)^3*(-1 + cot(a/4)^2)^6*sin(a/4)^12)

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