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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
          sin(x)            log(1 - cos(x))*sin(x)
------------------------- + ----------------------
(1 - cos(x))*(1 + cos(x))                   2     
                                (1 + cos(x))
log⁡(1−cos⁡(x))sin⁡(x)(cos⁡(x)+1)2+sin⁡(x)(1−cos⁡(x))(cos⁡(x)+1)\frac{\log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \frac{\sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)} 
sin(x)/(((1 - cos(x))*(1 + cos(x)))) + (log(1 - cos(x))*sin(x))/(1 + cos(x))^2
Simplificación general [src] 
1 - cos(x)*log(1 - cos(x)) + cos(x) + log(1 - cos(x))
-----------------------------------------------------
                 (1 + cos(x))*sin(x)
−log⁡(1−cos⁡(x))cos⁡(x)+log⁡(1−cos⁡(x))+cos⁡(x)+1(cos⁡(x)+1)sin⁡(x)\frac{- \log{\left(1 - \cos{\left(x \right)} \right)} \cos{\left(x \right)} + \log{\left(1 - \cos{\left(x \right)} \right)} + \cos{\left(x \right)} + 1}{\left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)}} 
(1 - cos(x)*log(1 - cos(x)) + cos(x) + log(1 - cos(x)))/((1 + cos(x))*sin(x))
Respuesta numérica [src] 
sin(x)/((1.0 - cos(x))*(1.0 + cos(x))) + log(1 - cos(x))*sin(x)/(1.0 + cos(x))^2
sin(x)/((1.0 - cos(x))*(1.0 + cos(x))) + log(1 - cos(x))*sin(x)/(1.0 + cos(x))^2
Unión de expresiones racionales [src] 
(1 + (1 - cos(x))*log(1 - cos(x)) + cos(x))*sin(x)
--------------------------------------------------
                                     2            
            (1 - cos(x))*(1 + cos(x))
((1−cos⁡(x))log⁡(1−cos⁡(x))+cos⁡(x)+1)sin⁡(x)(1−cos⁡(x))(cos⁡(x)+1)2\frac{\left(\left(1 - \cos{\left(x \right)}\right) \log{\left(1 - \cos{\left(x \right)} \right)} + \cos{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)^{2}} 
(1 + (1 - cos(x))*log(1 - cos(x)) + cos(x))*sin(x)/((1 - cos(x))*(1 + cos(x))^2)
Potencias [src] 
                                                                  /     I*x    -I*x\
                                              /   -I*x    I*x\    |    e      e    |
               /   -I*x    I*x\             I*\- e     + e   /*log|1 - ---- - -----|
             I*\- e     + e   /                                   \     2       2  /
- --------------------------------------- - ----------------------------------------
    /     I*x    -I*x\ /     I*x    -I*x\                                2          
    |    e      e    | |    e      e    |              /     I*x    -I*x\           
  2*|1 + ---- + -----|*|1 - ---- - -----|              |    e      e    |           
    \     2       2  / \     2       2  /            2*|1 + ---- + -----|           
                                                       \     2       2  /
−i(eix−e−ix)log⁡(−eix2+1−e−ix2)2(eix2+1+e−ix2)2−i(eix−e−ix)2(−eix2+1−e−ix2)(eix2+1+e−ix2)- \frac{i \left(e^{i x} - e^{- i x}\right) \log{\left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2} \right)}}{2 \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)^{2}} - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2}\right) \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)} 
-i*(-exp(-i*x) + exp(i*x))/(2*(1 + exp(i*x)/2 + exp(-i*x)/2)*(1 - exp(i*x)/2 - exp(-i*x)/2)) - i*(-exp(-i*x) + exp(i*x))*log(1 - exp(i*x)/2 - exp(-i*x)/2)/(2*(1 + exp(i*x)/2 + exp(-i*x)/2)^2)
Denominador racional [src] 
            2                                                          
(1 + cos(x)) *sin(x) + (1 - cos(x))*(1 + cos(x))*log(1 - cos(x))*sin(x)
-----------------------------------------------------------------------
                                                3                      
                       (1 - cos(x))*(1 + cos(x))
(1−cos⁡(x))(cos⁡(x)+1)log⁡(1−cos⁡(x))sin⁡(x)+(cos⁡(x)+1)2sin⁡(x)(1−cos⁡(x))(cos⁡(x)+1)3\frac{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right) \log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)^{3}} 
((1 + cos(x))^2*sin(x) + (1 - cos(x))*(1 + cos(x))*log(1 - cos(x))*sin(x))/((1 - cos(x))*(1 + cos(x))^3)
Combinatoria [src] 
(-1 - cos(x) - log(1 - cos(x)) + cos(x)*log(1 - cos(x)))*sin(x)
---------------------------------------------------------------
                              2                                
                  (1 + cos(x)) *(-1 + cos(x))
(log⁡(1−cos⁡(x))cos⁡(x)−log⁡(1−cos⁡(x))−cos⁡(x)−1)sin⁡(x)(cos⁡(x)−1)(cos⁡(x)+1)2\frac{\left(\log{\left(1 - \cos{\left(x \right)} \right)} \cos{\left(x \right)} - \log{\left(1 - \cos{\left(x \right)} \right)} - \cos{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)^{2}} 
(-1 - cos(x) - log(1 - cos(x)) + cos(x)*log(1 - cos(x)))*sin(x)/((1 + cos(x))^2*(-1 + cos(x)))
Abrimos la expresión [src] 
   sin(x)     log(1 - cos(x))*sin(x)
----------- + ----------------------
       2             2              
1 - cos (x)   1 + cos (x) + 2*cos(x)
log⁡(1−cos⁡(x))sin⁡(x)cos⁡2(x)+2cos⁡(x)+1+sin⁡(x)1−cos⁡2(x)\frac{\log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} + 2 \cos{\left(x \right)} + 1} + \frac{\sin{\left(x \right)}}{1 - \cos^{2}{\left(x \right)}} 
sin(x)/(1 - cos(x)^2) + log(1 - cos(x))*sin(x)/(1 + cos(x)^2 + 2*cos(x))
Parte trigonométrica [src] 
log(1 - cos(x))*sin(x)         
---------------------- + csc(x)
                2              
    (1 + cos(x))
csc⁡(x)+log⁡(1−cos⁡(x))sin⁡(x)(cos⁡(x)+1)2\csc{\left(x \right)} + \frac{\log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} 
                            /            2/x\\ 
                            |    -1 + cot |-|| 
                     /x\    |             \2/| 
                2*cot|-|*log|1 - ------------| 
       2/x\          \2/    |           2/x\ | 
1 + cot |-|                 |    1 + cot |-| | 
        \2/                 \            \2/ / 
----------- + ---------------------------------
       /x\                                    2
  2*cot|-|                  /            2/x\\ 
       \2/                  |    -1 + cot |-|| 
              /       2/x\\ |             \2/| 
              |1 + cot |-||*|1 + ------------| 
              \        \2// |           2/x\ | 
                            |    1 + cot |-| | 
                            \            \2/ /
cot⁡2(x2)+12cot⁡(x2)+2log⁡(−cot⁡2(x2)−1cot⁡2(x2)+1+1)cot⁡(x2)(cot⁡2(x2)−1cot⁡2(x2)+1+1)2(cot⁡2(x2)+1)\frac{\cot^{2}{\left(\frac{x}{2} \right)} + 1}{2 \cot{\left(\frac{x}{2} \right)}} + \frac{2 \log{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
      /         1     \           
   log|1 - -----------|           
      |       /pi    \|           
      |    csc|-- - x||           
      \       \2     //           
------------------------- + csc(x)
                 2                
/         1     \                 
|1 + -----------| *csc(x)         
|       /pi    \|                 
|    csc|-- - x||                 
\       \2     //
csc⁡(x)+log⁡(1−1csc⁡(−x+π2))(1+1csc⁡(−x+π2))2csc⁡(x)\csc{\left(x \right)} + \frac{\log{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}} 
  1      log(1 - cos(x))*sin(x)
------ + ----------------------
sin(x)                   2     
             (1 + cos(x))
1sin⁡(x)+log⁡(1−cos⁡(x))sin⁡(x)(cos⁡(x)+1)2\frac{1}{\sin{\left(x \right)}} + \frac{\log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} 
                                        /      1   \   
                                     log|1 - ------|   
               1                        \    sec(x)/   
-------------------------------- + --------------------
/      1   \ /      1   \                      2       
|1 + ------|*|1 - ------|*csc(x)   /      1   \        
\    sec(x)/ \    sec(x)/          |1 + ------| *csc(x)
                                   \    sec(x)/
log⁡(1−1sec⁡(x))(1+1sec⁡(x))2csc⁡(x)+1(1−1sec⁡(x))(1+1sec⁡(x))csc⁡(x)\frac{\log{\left(1 - \frac{1}{\sec{\left(x \right)}} \right)}}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right) \left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}} 
                                                /      1   \     
                                             log|1 - ------|     
                  1                             \    sec(x)/     
------------------------------------- + -------------------------
/      1   \ /      1   \    /    pi\               2            
|1 + ------|*|1 - ------|*sec|x - --|   /      1   \     /    pi\
\    sec(x)/ \    sec(x)/    \    2 /   |1 + ------| *sec|x - --|
                                        \    sec(x)/     \    2 /
log⁡(1−1sec⁡(x))(1+1sec⁡(x))2sec⁡(x−π2)+1(1−1sec⁡(x))(1+1sec⁡(x))sec⁡(x−π2)\frac{\log{\left(1 - \frac{1}{\sec{\left(x \right)}} \right)}}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)}}\right) \left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
          /    pi\             /    pi\                
       cos|x - --|          cos|x - --|*log(1 - cos(x))
          \    2 /             \    2 /                
------------------------- + ---------------------------
(1 - cos(x))*(1 + cos(x))                      2       
                                   (1 + cos(x))
log⁡(1−cos⁡(x))cos⁡(x−π2)(cos⁡(x)+1)2+cos⁡(x−π2)(1−cos⁡(x))(cos⁡(x)+1)\frac{\log{\left(1 - \cos{\left(x \right)} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)} 
                    /           2/x\\         
                    |    1 - tan |-||         
                    |            \2/|    /x\  
               2*log|1 - -----------|*tan|-|  
       2/x\         |           2/x\|    \2/  
1 + tan |-|         |    1 + tan |-||         
        \2/         \            \2//         
----------- + --------------------------------
       /x\                                   2
  2*tan|-|                  /           2/x\\ 
       \2/                  |    1 - tan |-|| 
              /       2/x\\ |            \2/| 
              |1 + tan |-||*|1 + -----------| 
              \        \2// |           2/x\| 
                            |    1 + tan |-|| 
                            \            \2//
tan⁡2(x2)+12tan⁡(x2)+2log⁡(−1−tan⁡2(x2)tan⁡2(x2)+1+1)tan⁡(x2)(1−tan⁡2(x2)tan⁡2(x2)+1+1)2(tan⁡2(x2)+1)\frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{2 \tan{\left(\frac{x}{2} \right)}} + \frac{2 \log{\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} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                                                                    /            2/x\\ 
                                                                    |    -1 + cot |-|| 
                                                             /x\    |             \2/| 
                                                        2*cot|-|*log|1 - ------------| 
                           /x\                               \2/    |           2/x\ | 
                      2*cot|-|                                      |    1 + cot |-| | 
                           \2/                                      \            \2/ / 
--------------------------------------------------- + ---------------------------------
              /            2/x\\ /            2/x\\                                   2
              |    -1 + cot |-|| |    -1 + cot |-||                 /            2/x\\ 
/       2/x\\ |             \2/| |             \2/|                 |    -1 + cot |-|| 
|1 + cot |-||*|1 + ------------|*|1 - ------------|   /       2/x\\ |             \2/| 
\        \2// |           2/x\ | |           2/x\ |   |1 + cot |-||*|1 + ------------| 
              |    1 + cot |-| | |    1 + cot |-| |   \        \2// |           2/x\ | 
              \            \2/ / \            \2/ /                 |    1 + cot |-| | 
                                                                    \            \2/ /
2log⁡(−cot⁡2(x2)−1cot⁡2(x2)+1+1)cot⁡(x2)(cot⁡2(x2)−1cot⁡2(x2)+1+1)2(cot⁡2(x2)+1)+2cot⁡(x2)(−cot⁡2(x2)−1cot⁡2(x2)+1+1)(cot⁡2(x2)−1cot⁡2(x2)+1+1)(cot⁡2(x2)+1)\frac{2 \log{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\frac{\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)} 
                 /    pi\                
              cos|x - --|*log(1 - cos(x))
     1           \    2 /                
----------- + ---------------------------
   /    pi\                      2       
cos|x - --|          (1 + cos(x))        
   \    2 /
1cos⁡(x−π2)+log⁡(1−cos⁡(x))cos⁡(x−π2)(cos⁡(x)+1)2\frac{1}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(1 - \cos{\left(x \right)} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x \right)} + 1\right)^{2}} 
                                         /       /    pi\\       
                                      log|1 - sin|x + --||*sin(x)
               sin(x)                    \       \    2 //       
----------------------------------- + ---------------------------
/       /    pi\\ /       /    pi\\                         2    
|1 - sin|x + --||*|1 + sin|x + --||        /       /    pi\\     
\       \    2 // \       \    2 //        |1 + sin|x + --||     
                                           \       \    2 //
log⁡(1−sin⁡(x+π2))sin⁡(x)(sin⁡(x+π2)+1)2+sin⁡(x)(1−sin⁡(x+π2))(sin⁡(x+π2)+1)\frac{\log{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)} \right)} \sin{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)^{2}} + \frac{\sin{\left(x \right)}}{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right) \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)} 
                                                   /         1     \  
                                                log|1 - -----------|  
                                                   |       /pi    \|  
                                                   |    csc|-- - x||  
                    1                              \       \2     //  
------------------------------------------ + -------------------------
/         1     \ /         1     \                           2       
|1 + -----------|*|1 - -----------|*csc(x)   /         1     \        
|       /pi    \| |       /pi    \|          |1 + -----------| *csc(x)
|    csc|-- - x|| |    csc|-- - x||          |       /pi    \|        
\       \2     // \       \2     //          |    csc|-- - x||        
                                             \       \2     //
log⁡(1−1csc⁡(−x+π2))(1+1csc⁡(−x+π2))2csc⁡(x)+1(1−1csc⁡(−x+π2))(1+1csc⁡(−x+π2))csc⁡(x)\frac{\log{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}} 
                                                          /           2/x\\         
                                                          |    1 - tan |-||         
                                                          |            \2/|    /x\  
                                                     2*log|1 - -----------|*tan|-|  
                          /x\                             |           2/x\|    \2/  
                     2*tan|-|                             |    1 + tan |-||         
                          \2/                             \            \2//         
------------------------------------------------- + --------------------------------
              /           2/x\\ /           2/x\\                                  2
              |    1 - tan |-|| |    1 - tan |-||                 /           2/x\\ 
/       2/x\\ |            \2/| |            \2/|                 |    1 - tan |-|| 
|1 + tan |-||*|1 + -----------|*|1 - -----------|   /       2/x\\ |            \2/| 
\        \2// |           2/x\| |           2/x\|   |1 + tan |-||*|1 + -----------| 
              |    1 + tan |-|| |    1 + tan |-||   \        \2// |           2/x\| 
              \            \2// \            \2//                 |    1 + tan |-|| 
                                                                  \            \2//
2log⁡(−1−tan⁡2(x2)tan⁡2(x2)+1+1)tan⁡(x2)(1−tan⁡2(x2)tan⁡2(x2)+1+1)2(tan⁡2(x2)+1)+2tan⁡(x2)(−1−tan⁡2(x2)tan⁡2(x2)+1+1)(1−tan⁡2(x2)tan⁡2(x2)+1+1)(tan⁡2(x2)+1)\frac{2 \log{\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} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
            /       /    pi\\       
         log|1 - sin|x + --||*sin(x)
  1         \       \    2 //       
------ + ---------------------------
sin(x)                         2    
              /       /    pi\\     
              |1 + sin|x + --||     
              \       \    2 //
1sin⁡(x)+log⁡(1−sin⁡(x+π2))sin⁡(x)(sin⁡(x+π2)+1)2\frac{1}{\sin{\left(x \right)}} + \frac{\log{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)} \right)} \sin{\left(x \right)}}{\left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)^{2}} 
        /      1   \                   
     log|1 - ------|                   
        \    sec(x)/           /    pi\
------------------------- + sec|x - --|
            2                  \    2 /
/      1   \     /    pi\              
|1 + ------| *sec|x - --|              
\    sec(x)/     \    2 /
sec⁡(x−π2)+log⁡(1−1sec⁡(x))(1+1sec⁡(x))2sec⁡(x−π2)\sec{\left(x - \frac{\pi}{2} \right)} + \frac{\log{\left(1 - \frac{1}{\sec{\left(x \right)}} \right)}}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}} 
log(1 - 1/sec(x))/((1 + 1/sec(x))^2*sec(x - pi/2)) + sec(x - pi/2)
Denominador común [src] 
-sin(x) - cos(x)*sin(x) - log(1 - cos(x))*sin(x) + cos(x)*log(1 - cos(x))*sin(x)
--------------------------------------------------------------------------------
                                2         3                                     
                        -1 + cos (x) + cos (x) - cos(x)
log⁡(1−cos⁡(x))sin⁡(x)cos⁡(x)−log⁡(1−cos⁡(x))sin⁡(x)−sin⁡(x)cos⁡(x)−sin⁡(x)cos⁡3(x)+cos⁡2(x)−cos⁡(x)−1\frac{\log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(1 - \cos{\left(x \right)} \right)} \sin{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)}}{\cos^{3}{\left(x \right)} + \cos^{2}{\left(x \right)} - \cos{\left(x \right)} - 1} 
(-sin(x) - cos(x)*sin(x) - log(1 - cos(x))*sin(x) + cos(x)*log(1 - cos(x))*sin(x))/(-1 + cos(x)^2 + cos(x)^3 - cos(x))
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