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

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

Solución

Ha introducido [src]

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

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

((cos(x)*(-sin(x) - cos(x)*sin(x)))/sqrt(1 + cos(x)^2))/(cos(x) + sqrt(1 + cos(x)^2))

Simplificación general [src]

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

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

-(1 + cos(x))*cos(x)*sin(x)/(1 + cos(x)^2 + sqrt(1 + cos(x)^2)*cos(x))

Respuesta numérica [src]

(1.0 + cos(x)^2)^(-0.5)*(-sin(x) - cos(x)*sin(x))*cos(x)/((1.0 + cos(x)^2)^0.5 + cos(x))

(1.0 + cos(x)^2)^(-0.5)*(-sin(x) - cos(x)*sin(x))*cos(x)/((1.0 + cos(x)^2)^0.5 + cos(x))

Combinatoria [src]

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

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

-(1 + cos(x))*cos(x)*sin(x)/(sqrt(1 + cos(x)^2)*(sqrt(1 + cos(x)^2) + cos(x)))

Denominador común [src]

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

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

-sin(x) + (-cos(x)*sin(x) + sqrt(1 + cos(x)^2)*cos(x)*sin(x) + sin(x))/(1 + cos(x)^2 + sqrt(1 + cos(x)^2)*cos(x))

Denominador racional [src]

                                     _____________                     _____________              
   2                3               /        2        2               /        2                  
cos (x)*sin(x) + cos (x)*sin(x) - \/  1 + cos (x) *cos (x)*sin(x) - \/  1 + cos (x) *cos(x)*sin(x)
--------------------------------------------------------------------------------------------------
                                            _____________                                         
                                           /        2                                             
                                         \/  1 + cos (x)

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

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

Potencias [src]

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

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

               /                       / I*x    -I*x\                 \ 
               |                       |e      e    | /   -I*x    I*x\| 
/ I*x    -I*x\ |  /   -I*x    I*x\   I*|---- + -----|*\- e     + e   /| 
|e      e    | |I*\- e     + e   /     \ 2       2  /                 | 
|---- + -----|*|------------------ + ---------------------------------| 
\ 2       2  / \        2                            2                / 
------------------------------------------------------------------------
      _____________________ /      _____________________               \
     /                   2  |     /                   2                |
    /      / I*x    -I*x\   |    /      / I*x    -I*x\      I*x    -I*x|
   /       |e      e    |   |   /       |e      e    |     e      e    |
  /    1 + |---- + -----|  *|  /    1 + |---- + -----|   + ---- + -----|
\/         \ 2       2  /   \\/         \ 2       2  /      2       2  /

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

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

Unión de expresiones racionales [src]

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

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

(-1 - cos(x))*cos(x)*sin(x)/(sqrt(1 + cos(x)^2)*(sqrt(1 + cos(x)^2) + cos(x)))

Compilar la expresión [src]

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

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

(-sin(x) - cos(x)*sin(x))*cos(x)/(sqrt(1 + cos(x)^2)*(sqrt(1 + cos(x)^2) + cos(x)))

Abrimos la expresión [src]

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

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

-cos(x)^2*sin(x)/(1 + cos(x)^2 + sqrt(1 + cos(x)^2)*cos(x)) - cos(x)*sin(x)/(1 + cos(x)^2 + sqrt(1 + cos(x)^2)*cos(x))

Parte trigonométrica [src]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

                    1            1                     
                - ------ - -------------               
                  csc(x)   csc(x)*sec(x)               
-------------------------------------------------------
     _____________ /     _____________         \       
    /        1     |    /        1         1   |       
   /  1 + ------- *|   /  1 + -------  + ------|*sec(x)
  /          2     |  /          2       sec(x)|       
\/        sec (x)  \\/        sec (x)          /

\frac{- \frac{1}{\csc{\left(x \right)}} - \frac{1}{\csc{\left(x \right)} \sec{\left(x \right)}}}{\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} \left(\sqrt{1 + \frac{1}{\sec^{2}{\left(x \right)}}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}}

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

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

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

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

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

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

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

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

      2/x\    2/x\    4/x\              /        2/x\\    /x\
-8*cot |-|*sin |-|*sin |-|*(1 + cos(x))*|-1 + cot |-||*cot|-|
       \4/     \2/     \4/              \         \2//    \2/
-------------------------------------------------------------
            _____________ /   _____________         \        
           /        2     |  /        2             |        
         \/  1 + cos (x) *\\/  1 + cos (x)  + cos(x)/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(-sin(x) - cos(x)*sin(x))*cos(x)/(sqrt(1 + cos(x)^2)*(sqrt(1 + cos(x)^2) + cos(x)))

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

Solución

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¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

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

Solución

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