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Simplificación de expresiones/ Descomposición en fracciones simples/ sin^2a(cos4a-cos2a)/(sina-2sin3a+sin5a)

¿Cómo vas a descomponer esta sin^2a(cos4a-cos2a)/(sina-2sin3a+sin5a) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   2                          
sin (a)*(cos(4*a) - cos(2*a)) 
------------------------------
sin(a) - 2*sin(3*a) + sin(5*a)
$$\frac{\left(- \cos{\left(2 a \right)} + \cos{\left(4 a \right)}\right) \sin^{2}{\left(a \right)}}{\left(\sin{\left(a \right)} - 2 \sin{\left(3 a \right)}\right) + \sin{\left(5 a \right)}}$$ 
(sin(a)^2*(cos(4*a) - cos(2*a)))/(sin(a) - 2*sin(3*a) + sin(5*a))
Simplificación general [src] 
       4    /         2   \    
    sin (a)*\6 - 8*sin (a)/    
-------------------------------
-sin(a) - sin(5*a) + 2*sin(3*a)
$$\frac{\left(6 - 8 \sin^{2}{\left(a \right)}\right) \sin^{4}{\left(a \right)}}{- \sin{\left(a \right)} + 2 \sin{\left(3 a \right)} - \sin{\left(5 a \right)}}$$ 
sin(a)^4*(6 - 8*sin(a)^2)/(-sin(a) - sin(5*a) + 2*sin(3*a))
Respuesta numérica [src] 
sin(a)^2*(-cos(2*a) + cos(4*a))/(-2.0*sin(3*a) + sin(a) + sin(5*a))
sin(a)^2*(-cos(2*a) + cos(4*a))/(-2.0*sin(3*a) + sin(a) + sin(5*a))
Denominador racional [src] 
    2                          
 sin (a)*(-cos(2*a) + cos(4*a))
-------------------------------
-2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{\left(- \cos{\left(2 a \right)} + \cos{\left(4 a \right)}\right) \sin^{2}{\left(a \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
sin(a)^2*(-cos(2*a) + cos(4*a))/(-2*sin(3*a) + sin(a) + sin(5*a))
Unión de expresiones racionales [src] 
    2                          
 sin (a)*(-cos(2*a) + cos(4*a))
-------------------------------
-2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{\left(- \cos{\left(2 a \right)} + \cos{\left(4 a \right)}\right) \sin^{2}{\left(a \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
sin(a)^2*(-cos(2*a) + cos(4*a))/(-2*sin(3*a) + sin(a) + sin(5*a))
Potencias [src] 
    2                          
 sin (a)*(-cos(2*a) + cos(4*a))
-------------------------------
-2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{\left(- \cos{\left(2 a \right)} + \cos{\left(4 a \right)}\right) \sin^{2}{\left(a \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
                        2 / -4*I*a    4*I*a    -2*I*a    2*I*a\         
        /   -I*a    I*a\  |e         e        e         e     |         
       -\- e     + e   / *|------- + ------ - ------- - ------|         
                          \   2        2         2        2   /         
------------------------------------------------------------------------
  /                           /   -I*a    I*a\     /   -5*I*a    5*I*a\\
  |  /   -3*I*a    3*I*a\   I*\- e     + e   /   I*\- e       + e     /|
4*|I*\- e       + e     / - ------------------ - ----------------------|
  \                                 2                      2           /
$$- \frac{\left(e^{i a} - e^{- i a}\right)^{2} \left(\frac{e^{4 i a}}{2} - \frac{e^{2 i a}}{2} - \frac{e^{- 2 i a}}{2} + \frac{e^{- 4 i a}}{2}\right)}{4 \left(- \frac{i \left(e^{i a} - e^{- i a}\right)}{2} + i \left(e^{3 i a} - e^{- 3 i a}\right) - \frac{i \left(e^{5 i a} - e^{- 5 i a}\right)}{2}\right)}$$ 
-(-exp(-i*a) + exp(i*a))^2*(exp(-4*i*a)/2 + exp(4*i*a)/2 - exp(-2*i*a)/2 - exp(2*i*a)/2)/(4*(i*(-exp(-3*i*a) + exp(3*i*a)) - i*(-exp(-i*a) + exp(i*a))/2 - i*(-exp(-5*i*a) + exp(5*i*a))/2))
Denominador común [src] 
   2                  2            
sin (a)*cos(2*a) - sin (a)*cos(4*a)
-----------------------------------
  -sin(a) - sin(5*a) + 2*sin(3*a)
$$\frac{\sin^{2}{\left(a \right)} \cos{\left(2 a \right)} - \sin^{2}{\left(a \right)} \cos{\left(4 a \right)}}{- \sin{\left(a \right)} + 2 \sin{\left(3 a \right)} - \sin{\left(5 a \right)}}$$ 
(sin(a)^2*cos(2*a) - sin(a)^2*cos(4*a))/(-sin(a) - sin(5*a) + 2*sin(3*a))
Combinatoria [src] 
    2                          
 sin (a)*(-cos(4*a) + cos(2*a))
-------------------------------
-sin(a) - sin(5*a) + 2*sin(3*a)
$$\frac{\left(\cos{\left(2 a \right)} - \cos{\left(4 a \right)}\right) \sin^{2}{\left(a \right)}}{- \sin{\left(a \right)} + 2 \sin{\left(3 a \right)} - \sin{\left(5 a \right)}}$$ 
sin(a)^2*(-cos(4*a) + cos(2*a))/(-sin(a) - sin(5*a) + 2*sin(3*a))
Compilar la expresión [src] 
    2                          
 sin (a)*(-cos(2*a) + cos(4*a))
-------------------------------
-2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{\left(- \cos{\left(2 a \right)} + \cos{\left(4 a \right)}\right) \sin^{2}{\left(a \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
sin(a)^2*(-cos(2*a) + cos(4*a))/(-2*sin(3*a) + sin(a) + sin(5*a))
Abrimos la expresión [src] 
             2                        2       2                  4       2       
        2*sin (a)               10*cos (a)*sin (a)          8*cos (a)*sin (a)    
------------------------- - ------------------------- + -------------------------
        3            5              3            5              3            5   
- 12*sin (a) + 16*sin (a)   - 12*sin (a) + 16*sin (a)   - 12*sin (a) + 16*sin (a)
$$\frac{8 \sin^{2}{\left(a \right)} \cos^{4}{\left(a \right)}}{16 \sin^{5}{\left(a \right)} - 12 \sin^{3}{\left(a \right)}} - \frac{10 \sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)}}{16 \sin^{5}{\left(a \right)} - 12 \sin^{3}{\left(a \right)}} + \frac{2 \sin^{2}{\left(a \right)}}{16 \sin^{5}{\left(a \right)} - 12 \sin^{3}{\left(a \right)}}$$ 
2*sin(a)^2/(-12*sin(a)^3 + 16*sin(a)^5) - 10*cos(a)^2*sin(a)^2/(-12*sin(a)^3 + 16*sin(a)^5) + 8*cos(a)^4*sin(a)^2/(-12*sin(a)^3 + 16*sin(a)^5)
Parte trigonométrica [src] 
                    /        2                2   \           
               2/a\ |-1 + cot (2*a)   -1 + cot (a)|           
          4*cot |-|*|-------------- - ------------|           
                \2/ |       2                2    |           
                    \1 + cot (2*a)    1 + cot (a) /           
--------------------------------------------------------------
               /         /3*a\           /a\           /5*a\ \
             2 |    4*cot|---|      2*cot|-|      2*cot|---| |
/       2/a\\  |         \ 2 /           \2/           \ 2 / |
|1 + cot |-|| *|- ------------- + ----------- + -------------|
\        \2//  |         2/3*a\          2/a\          2/5*a\|
               |  1 + cot |---|   1 + cot |-|   1 + cot |---||
               \          \ 2 /           \2/           \ 2 //
$$\frac{4 \left(- \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + \frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1}\right) \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\frac{2 \cot{\left(\frac{5 a}{2} \right)}}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} - \frac{4 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right)}$$ 
           4/    pi\ /         2/    pi\\      
      2*cos |a - --|*|3 - 4*cos |a - --||      
            \    2 / \          \    2 //      
-----------------------------------------------
     /    pi\      /      pi\        /      pi\
- cos|a - --| - cos|5*a - --| + 2*cos|3*a - --|
     \    2 /      \      2 /        \      2 /
$$\frac{2 \left(3 - 4 \cos^{2}{\left(a - \frac{\pi}{2} \right)}\right) \cos^{4}{\left(a - \frac{\pi}{2} \right)}}{- \cos{\left(a - \frac{\pi}{2} \right)} + 2 \cos{\left(3 a - \frac{\pi}{2} \right)} - \cos{\left(5 a - \frac{\pi}{2} \right)}}$$ 
    2                          
 sin (a)*(-cos(2*a) + cos(4*a))
-------------------------------
-2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{\left(- \cos{\left(2 a \right)} + \cos{\left(4 a \right)}\right) \sin^{2}{\left(a \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
   2    /       2           4   \
sin (a)*\- 6*sin (a) + 8*sin (a)/
---------------------------------
 -2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{\left(8 \sin^{4}{\left(a \right)} - 6 \sin^{2}{\left(a \right)}\right) \sin^{2}{\left(a \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
                          /            2/a\  \                
                          |      16*tan |-|  |                
                     4/a\ |             \2/  |                
               32*tan |-|*|3 - --------------|                
                      \2/ |                 2|                
                          |    /       2/a\\ |                
                          |    |1 + tan |-|| |                
                          \    \        \2// /                
--------------------------------------------------------------
               /         /a\           /5*a\           /3*a\ \
             4 |    2*tan|-|      2*tan|---|      4*tan|---| |
/       2/a\\  |         \2/           \ 2 /           \ 2 / |
|1 + tan |-|| *|- ----------- - ------------- + -------------|
\        \2//  |         2/a\          2/5*a\          2/3*a\|
               |  1 + tan |-|   1 + tan |---|   1 + tan |---||
               \          \2/           \ 2 /           \ 2 //
$$\frac{32 \left(3 - \frac{16 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}\right) \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \left(- \frac{2 \tan{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1} + \frac{4 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right)}$$ 
     4/a\    2/a\ /       2           4   \
4*cos |-|*tan |-|*\- 6*sin (a) + 8*sin (a)/
      \2/     \2/                          
-------------------------------------------
      -2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{4 \left(8 \sin^{4}{\left(a \right)} - 6 \sin^{2}{\left(a \right)}\right) \cos^{4}{\left(\frac{a}{2} \right)} \tan^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
                      1          1                        
                   -------- - --------                    
                   sec(4*a)   sec(2*a)                    
----------------------------------------------------------
/     1              1               2      \    2/    pi\
|----------- + ------------- - -------------|*sec |a - --|
|   /    pi\      /      pi\      /      pi\|     \    2 /
|sec|a - --|   sec|5*a - --|   sec|3*a - --||             
\   \    2 /      \      2 /      \      2 //
$$\frac{\frac{1}{\sec{\left(4 a \right)}} - \frac{1}{\sec{\left(2 a \right)}}}{\left(\frac{1}{\sec{\left(5 a - \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$ 
              /       4   \             
            2*|3 - -------|             
              |       2   |             
              \    csc (a)/             
----------------------------------------
/    1         1          2    \    4   
|- ------ - -------- + --------|*csc (a)
\  csc(a)   csc(5*a)   csc(3*a)/
$$\frac{2 \left(3 - \frac{4}{\csc^{2}{\left(a \right)}}\right)}{\left(- \frac{1}{\csc{\left(5 a \right)}} + \frac{2}{\csc{\left(3 a \right)}} - \frac{1}{\csc{\left(a \right)}}\right) \csc^{4}{\left(a \right)}}$$ 
          1               1           
    ------------- - -------------     
       /pi      \      /pi      \     
    csc|-- - 4*a|   csc|-- - 2*a|     
       \2       /      \2       /     
--------------------------------------
/  1         1          2    \    2   
|------ + -------- - --------|*csc (a)
\csc(a)   csc(5*a)   csc(3*a)/
$$\frac{- \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}}{\left(\frac{1}{\csc{\left(5 a \right)}} - \frac{2}{\csc{\left(3 a \right)}} + \frac{1}{\csc{\left(a \right)}}\right) \csc^{2}{\left(a \right)}}$$ 
                       /          2/a\             4/a\  \            
                       |    24*tan |-|      128*tan |-|  |            
                  4/a\ |           \2/              \2/  |            
            64*tan |-|*|- -------------- + --------------|            
                   \4/ |               2                4|            
                       |  /       2/a\\    /       2/a\\ |            
                       |  |1 + tan |-||    |1 + tan |-|| |            
                       \  \        \2//    \        \2// /            
----------------------------------------------------------------------
               /         /3*a\           /a\           /5*a\ \        
             4 |    4*tan|---|      2*tan|-|      2*tan|---| |        
/       2/a\\  |         \ 2 /           \2/           \ 2 / |    2/a\
|1 + tan |-|| *|- ------------- + ----------- + -------------|*tan |-|
\        \4//  |         2/3*a\          2/a\          2/5*a\|     \2/
               |  1 + tan |---|   1 + tan |-|   1 + tan |---||        
               \          \ 2 /           \2/           \ 2 //
$$\frac{64 \left(- \frac{24 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{128 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}\right) \tan^{4}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4} \left(\frac{2 \tan{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1} - \frac{4 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \tan^{2}{\left(\frac{a}{2} \right)}}$$ 
     2/a\    2/a\ /       2           4   \
4*cos |-|*sin |-|*\- 6*sin (a) + 8*sin (a)/
      \2/     \2/                          
-------------------------------------------
      -2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{4 \left(8 \sin^{4}{\left(a \right)} - 6 \sin^{2}{\left(a \right)}\right) \sin^{2}{\left(\frac{a}{2} \right)} \cos^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
         2/    pi\                             
      cos |a - --|*(-cos(2*a) + cos(4*a))      
          \    2 /                             
-----------------------------------------------
       /      pi\      /    pi\      /      pi\
- 2*cos|3*a - --| + cos|a - --| + cos|5*a - --|
       \      2 /      \    2 /      \      2 /
$$\frac{\left(- \cos{\left(2 a \right)} + \cos{\left(4 a \right)}\right) \cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a - \frac{\pi}{2} \right)} - 2 \cos{\left(3 a - \frac{\pi}{2} \right)} + \cos{\left(5 a - \frac{\pi}{2} \right)}}$$ 
      4/a\    8/a\ /         2   \
32*cot |-|*sin |-|*\3 - 4*sin (a)/
       \2/     \2/                
----------------------------------
 -sin(a) - sin(5*a) + 2*sin(3*a)
$$\frac{32 \left(3 - 4 \sin^{2}{\left(a \right)}\right) \sin^{8}{\left(\frac{a}{2} \right)} \cot^{4}{\left(\frac{a}{2} \right)}}{- \sin{\left(a \right)} + 2 \sin{\left(3 a \right)} - \sin{\left(5 a \right)}}$$ 
   2    /     /pi      \      /pi      \\
sin (a)*|- sin|-- + 2*a| + sin|-- + 4*a||
        \     \2       /      \2       //
-----------------------------------------
     -2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{\left(- \sin{\left(2 a + \frac{\pi}{2} \right)} + \sin{\left(4 a + \frac{\pi}{2} \right)}\right) \sin^{2}{\left(a \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
            1          1              
         -------- - --------          
         sec(4*a)   sec(2*a)          
--------------------------------------
/  1         1          2    \    2   
|------ + -------- - --------|*csc (a)
\csc(a)   csc(5*a)   csc(3*a)/
$$\frac{\frac{1}{\sec{\left(4 a \right)}} - \frac{1}{\sec{\left(2 a \right)}}}{\left(\frac{1}{\csc{\left(5 a \right)}} - \frac{2}{\csc{\left(3 a \right)}} + \frac{1}{\csc{\left(a \right)}}\right) \csc^{2}{\left(a \right)}}$$ 
                       /          2/a\             4/a\  \    
                       |    24*cot |-|      128*cot |-|  |    
          2/a\    4/a\ |           \2/              \2/  |    
    64*cot |-|*cot |-|*|- -------------- + --------------|    
           \2/     \4/ |               2                4|    
                       |  /       2/a\\    /       2/a\\ |    
                       |  |1 + cot |-||    |1 + cot |-|| |    
                       \  \        \2//    \        \2// /    
--------------------------------------------------------------
               /         /3*a\           /a\           /5*a\ \
             4 |    4*cot|---|      2*cot|-|      2*cot|---| |
/       2/a\\  |         \ 2 /           \2/           \ 2 / |
|1 + cot |-|| *|- ------------- + ----------- + -------------|
\        \4//  |         2/3*a\          2/a\          2/5*a\|
               |  1 + cot |---|   1 + cot |-|   1 + cot |---||
               \          \ 2 /           \2/           \ 2 //
$$\frac{64 \left(- \frac{24 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{128 \cot^{4}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}\right) \cot^{4}{\left(\frac{a}{4} \right)} \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4} \left(\frac{2 \cot{\left(\frac{5 a}{2} \right)}}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} - \frac{4 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right)}$$ 
               4                                                          
  /       2/a\\     8/a\    2/a\ /                           8/a\    4/a\\
4*|1 - tan |-|| *cos |-|*tan |-|*|-3*(1 - cos(2*a)) + 128*cos |-|*tan |-||
  \        \4//      \4/     \2/ \                            \2/     \2//
--------------------------------------------------------------------------
                     -2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{4} \left(- 3 \left(1 - \cos{\left(2 a \right)}\right) + 128 \cos^{8}{\left(\frac{a}{2} \right)} \tan^{4}{\left(\frac{a}{2} \right)}\right) \cos^{8}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
                     /       2               2   \            
                2/a\ |1 - tan (2*a)   1 - tan (a)|            
           4*tan |-|*|------------- - -----------|            
                 \2/ |       2               2   |            
                     \1 + tan (2*a)   1 + tan (a)/            
--------------------------------------------------------------
               /         /3*a\           /a\           /5*a\ \
             2 |    4*tan|---|      2*tan|-|      2*tan|---| |
/       2/a\\  |         \ 2 /           \2/           \ 2 / |
|1 + tan |-|| *|- ------------- + ----------- + -------------|
\        \2//  |         2/3*a\          2/a\          2/5*a\|
               |  1 + tan |---|   1 + tan |-|   1 + tan |---||
               \          \ 2 /           \2/           \ 2 //
$$\frac{4 \left(- \frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + \frac{1 - \tan^{2}{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}\right) \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\frac{2 \tan{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1} - \frac{4 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right)}$$ 
                          /            2/a\  \                
                          |      16*cot |-|  |                
                     4/a\ |             \2/  |                
               32*cot |-|*|3 - --------------|                
                      \2/ |                 2|                
                          |    /       2/a\\ |                
                          |    |1 + cot |-|| |                
                          \    \        \2// /                
--------------------------------------------------------------
               /         /a\           /5*a\           /3*a\ \
             4 |    2*cot|-|      2*cot|---|      4*cot|---| |
/       2/a\\  |         \2/           \ 2 /           \ 2 / |
|1 + cot |-|| *|- ----------- - ------------- + -------------|
\        \2//  |         2/a\          2/5*a\          2/3*a\|
               |  1 + cot |-|   1 + cot |---|   1 + cot |---||
               \          \2/           \ 2 /           \ 2 //
$$\frac{32 \left(3 - \frac{16 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}\right) \cot^{4}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4} \left(- \frac{2 \cot{\left(\frac{5 a}{2} \right)}}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} + \frac{4 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right)}$$ 
                  /       6              8      \                 
                4*|- ------------ + ------------|                 
                  |     2/    pi\      4/    pi\|                 
                  |  sec |a - --|   sec |a - --||                 
                  \      \    2 /       \    2 //                 
------------------------------------------------------------------
/     1              1               2      \    2/a\    2/a   pi\
|----------- + ------------- - -------------|*sec |-|*sec |- - --|
|   /    pi\      /      pi\      /      pi\|     \2/     \2   2 /
|sec|a - --|   sec|5*a - --|   sec|3*a - --||                     
\   \    2 /      \      2 /      \      2 //
$$\frac{4 \left(- \frac{6}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + \frac{8}{\sec^{4}{\left(a - \frac{\pi}{2} \right)}}\right)}{\left(\frac{1}{\sec{\left(5 a - \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(\frac{a}{2} \right)} \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$ 
      4/a\    4/a\ /         2   \
32*cos |-|*sin |-|*\3 - 4*sin (a)/
       \2/     \2/                
----------------------------------
 -sin(a) - sin(5*a) + 2*sin(3*a)
$$\frac{32 \left(3 - 4 \sin^{2}{\left(a \right)}\right) \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)}}{- \sin{\left(a \right)} + 2 \sin{\left(3 a \right)} - \sin{\left(5 a \right)}}$$ 
                          /          2/a\             4/a\  \         
                        4 |    24*cot |-|      128*cot |-|  |         
          /        2/a\\  |           \2/              \2/  |         
        4*|-1 + cot |-|| *|- -------------- + --------------|         
          \         \4//  |               2                4|         
                          |  /       2/a\\    /       2/a\\ |         
                          |  |1 + cot |-||    |1 + cot |-|| |         
                          \  \        \2//    \        \2// /         
----------------------------------------------------------------------
               /         /3*a\           /a\           /5*a\ \        
             4 |    4*cot|---|      2*cot|-|      2*cot|---| |        
/       2/a\\  |         \ 2 /           \2/           \ 2 / |    2/a\
|1 + cot |-|| *|- ------------- + ----------- + -------------|*cot |-|
\        \4//  |         2/3*a\          2/a\          2/5*a\|     \2/
               |  1 + cot |---|   1 + cot |-|   1 + cot |---||        
               \          \ 2 /           \2/           \ 2 //
$$\frac{4 \left(- \frac{24 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{128 \cot^{4}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4} \left(\frac{2 \cot{\left(\frac{5 a}{2} \right)}}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} - \frac{4 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \cot^{2}{\left(\frac{a}{2} \right)}}$$ 
      2/a\    4/a\    8/a\ /                           4/a\    8/a\\
64*cot |-|*cot |-|*sin |-|*|-3*(1 - cos(2*a)) + 128*cot |-|*sin |-||
       \2/     \4/     \4/ \                            \2/     \2//
--------------------------------------------------------------------
                  -2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{64 \left(- 3 \left(1 - \cos{\left(2 a \right)}\right) + 128 \sin^{8}{\left(\frac{a}{2} \right)} \cot^{4}{\left(\frac{a}{2} \right)}\right) \sin^{8}{\left(\frac{a}{4} \right)} \cot^{4}{\left(\frac{a}{4} \right)} \cot^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
           4           6       
    - 6*sin (a) + 8*sin (a)    
-------------------------------
-2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{8 \sin^{6}{\left(a \right)} - 6 \sin^{4}{\left(a \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
      4/a\    4/pi   a\ /       2           4   \
16*sin |-|*sin |-- + -|*\- 6*sin (a) + 8*sin (a)/
       \2/     \2    2/                          
-------------------------------------------------
                                         2       
    (-2*sin(3*a) + sin(a) + sin(5*a))*sin (a)
$$\frac{16 \left(8 \sin^{4}{\left(a \right)} - 6 \sin^{2}{\left(a \right)}\right) \sin^{4}{\left(\frac{a}{2} \right)} \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{\left(\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}\right) \sin^{2}{\left(a \right)}}$$ 
      8/a\    4/a\ /         2   \
32*cos |-|*tan |-|*\3 - 4*sin (a)/
       \2/     \2/                
----------------------------------
 -sin(a) - sin(5*a) + 2*sin(3*a)
$$\frac{32 \left(3 - 4 \sin^{2}{\left(a \right)}\right) \cos^{8}{\left(\frac{a}{2} \right)} \tan^{4}{\left(\frac{a}{2} \right)}}{- \sin{\left(a \right)} + 2 \sin{\left(3 a \right)} - \sin{\left(5 a \right)}}$$ 
     2/a\    2/a   pi\ /       2/    pi\        4/    pi\\
4*cos |-|*cos |- - --|*|- 6*cos |a - --| + 8*cos |a - --||
      \2/     \2   2 / \        \    2 /         \    2 //
----------------------------------------------------------
            /      pi\      /    pi\      /      pi\      
     - 2*cos|3*a - --| + cos|a - --| + cos|5*a - --|      
            \      2 /      \    2 /      \      2 /
$$\frac{4 \left(8 \cos^{4}{\left(a - \frac{\pi}{2} \right)} - 6 \cos^{2}{\left(a - \frac{\pi}{2} \right)}\right) \cos^{2}{\left(\frac{a}{2} \right)} \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos{\left(a - \frac{\pi}{2} \right)} - 2 \cos{\left(3 a - \frac{\pi}{2} \right)} + \cos{\left(5 a - \frac{\pi}{2} \right)}}$$ 
     2/a\    4/a\ /       2           4   \
4*cot |-|*sin |-|*\- 6*sin (a) + 8*sin (a)/
      \2/     \2/                          
-------------------------------------------
      -2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{4 \left(8 \sin^{4}{\left(a \right)} - 6 \sin^{2}{\left(a \right)}\right) \sin^{4}{\left(\frac{a}{2} \right)} \cot^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
        4    /         2   \   
   2*sin (a)*\3 - 4*sin (a)/   
-------------------------------
-sin(a) - sin(5*a) + 2*sin(3*a)
$$\frac{2 \left(3 - 4 \sin^{2}{\left(a \right)}\right) \sin^{4}{\left(a \right)}}{- \sin{\left(a \right)} + 2 \sin{\left(3 a \right)} - \sin{\left(5 a \right)}}$$ 
                      /         4      \                    
                    2*|3 - ------------|                    
                      |       2/    pi\|                    
                      |    sec |a - --||                    
                      \        \    2 //                    
------------------------------------------------------------
/       1              1               2      \    4/    pi\
|- ----------- - ------------- + -------------|*sec |a - --|
|     /    pi\      /      pi\      /      pi\|     \    2 /
|  sec|a - --|   sec|5*a - --|   sec|3*a - --||             
\     \    2 /      \      2 /      \      2 //
$$\frac{2 \left(3 - \frac{4}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}\right)}{\left(- \frac{1}{\sec{\left(5 a - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(3 a - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec^{4}{\left(a - \frac{\pi}{2} \right)}}$$ 
      8/a\    4/a\ /                           8/a\    4/a\\
64*cos |-|*tan |-|*|-3*(1 - cos(2*a)) + 128*cos |-|*tan |-||
       \4/     \4/ \                            \2/     \2//
------------------------------------------------------------
                                              2/a\          
         (-2*sin(3*a) + sin(a) + sin(5*a))*tan |-|          
                                               \2/
$$\frac{64 \left(- 3 \left(1 - \cos{\left(2 a \right)}\right) + 128 \cos^{8}{\left(\frac{a}{2} \right)} \tan^{4}{\left(\frac{a}{2} \right)}\right) \cos^{8}{\left(\frac{a}{4} \right)} \tan^{4}{\left(\frac{a}{4} \right)}}{\left(\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}\right) \tan^{2}{\left(\frac{a}{2} \right)}}$$ 
                /     6         8   \              
              4*|- ------- + -------|              
                |     2         4   |              
                \  csc (a)   csc (a)/              
---------------------------------------------------
/  1         1          2    \    2/a\    2/pi   a\
|------ + -------- - --------|*csc |-|*csc |-- - -|
\csc(a)   csc(5*a)   csc(3*a)/     \2/     \2    2/
$$\frac{4 \left(- \frac{6}{\csc^{2}{\left(a \right)}} + \frac{8}{\csc^{4}{\left(a \right)}}\right)}{\left(\frac{1}{\csc{\left(5 a \right)}} - \frac{2}{\csc{\left(3 a \right)}} + \frac{1}{\csc{\left(a \right)}}\right) \csc^{2}{\left(\frac{a}{2} \right)} \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$ 
                4                                                          
  /        2/a\\     8/a\    2/a\ /                           4/a\    8/a\\
4*|-1 + cot |-|| *sin |-|*tan |-|*|-3*(1 - cos(2*a)) + 128*cot |-|*sin |-||
  \         \4//      \4/     \2/ \                            \2/     \2//
---------------------------------------------------------------------------
                      -2*sin(3*a) + sin(a) + sin(5*a)
$$\frac{4 \left(- 3 \left(1 - \cos{\left(2 a \right)}\right) + 128 \sin^{8}{\left(\frac{a}{2} \right)} \cot^{4}{\left(\frac{a}{2} \right)}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)} - 2 \sin{\left(3 a \right)} + \sin{\left(5 a \right)}}$$ 
                          /          2/a\             4/a\  \ 
                4         |    24*tan |-|      128*tan |-|  | 
   /       2/a\\     2/a\ |           \2/              \2/  | 
 4*|1 - tan |-|| *tan |-|*|- -------------- + --------------| 
   \        \4//      \2/ |               2                4| 
                          |  /       2/a\\    /       2/a\\ | 
                          |  |1 + tan |-||    |1 + tan |-|| | 
                          \  \        \2//    \        \2// / 
--------------------------------------------------------------
               /         /3*a\           /a\           /5*a\ \
             4 |    4*tan|---|      2*tan|-|      2*tan|---| |
/       2/a\\  |         \ 2 /           \2/           \ 2 / |
|1 + tan |-|| *|- ------------- + ----------- + -------------|
\        \4//  |         2/3*a\          2/a\          2/5*a\|
               |  1 + tan |---|   1 + tan |-|   1 + tan |---||
               \          \ 2 /           \2/           \ 2 //
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{4} \left(- \frac{24 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{128 \tan^{4}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{4}}\right) \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4} \left(\frac{2 \tan{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1} - \frac{4 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right)}$$ 
4*(1 - tan(a/4)^2)^4*tan(a/2)^2*(-24*tan(a/2)^2/(1 + tan(a/2)^2)^2 + 128*tan(a/2)^4/(1 + tan(a/2)^2)^4)/((1 + tan(a/4)^2)^4*(-4*tan(3*a/2)/(1 + tan(3*a/2)^2) + 2*tan(a/2)/(1 + tan(a/2)^2) + 2*tan(5*a/2)/(1 + tan(5*a/2)^2)))
    © Sr Examen