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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    3             3     
 sin (x)       sin (x)  
---------- + -----------
1 + cos(x)          2   
             1 - sin (x)
sin⁡3(x)cos⁡(x)+1+sin⁡3(x)1−sin⁡2(x)\frac{\sin^{3}{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\sin^{3}{\left(x \right)}}{1 - \sin^{2}{\left(x \right)}} 
sin(x)^3/(1 + cos(x)) + sin(x)^3/(1 - sin(x)^2)
Simplificación general [src] 
   3    /       2            \
sin (x)*\2 - sin (x) + cos(x)/
------------------------------
                     2        
     (1 + cos(x))*cos (x)
(−sin⁡2(x)+cos⁡(x)+2)sin⁡3(x)(cos⁡(x)+1)cos⁡2(x)\frac{\left(- \sin^{2}{\left(x \right)} + \cos{\left(x \right)} + 2\right) \sin^{3}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}} 
sin(x)^3*(2 - sin(x)^2 + cos(x))/((1 + cos(x))*cos(x)^2)
Respuesta numérica [src] 
sin(x)^3/(1.0 - sin(x)^2) + sin(x)^3/(1.0 + cos(x))
sin(x)^3/(1.0 - sin(x)^2) + sin(x)^3/(1.0 + cos(x))
Denominador racional [src] 
   3    /       2   \      3                
sin (x)*\1 - sin (x)/ + sin (x)*(1 + cos(x))
--------------------------------------------
         /       2   \                      
         \1 - sin (x)/*(1 + cos(x))
(1−sin⁡2(x))sin⁡3(x)+(cos⁡(x)+1)sin⁡3(x)(1−sin⁡2(x))(cos⁡(x)+1)\frac{\left(1 - \sin^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right) \sin^{3}{\left(x \right)}}{\left(1 - \sin^{2}{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)} 
(sin(x)^3*(1 - sin(x)^2) + sin(x)^3*(1 + cos(x)))/((1 - sin(x)^2)*(1 + cos(x)))
Unión de expresiones racionales [src] 
   3    /       2            \
sin (x)*\2 - sin (x) + cos(x)/
------------------------------
  /       2   \               
  \1 - sin (x)/*(1 + cos(x))
(−sin⁡2(x)+cos⁡(x)+2)sin⁡3(x)(1−sin⁡2(x))(cos⁡(x)+1)\frac{\left(- \sin^{2}{\left(x \right)} + \cos{\left(x \right)} + 2\right) \sin^{3}{\left(x \right)}}{\left(1 - \sin^{2}{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)} 
sin(x)^3*(2 - sin(x)^2 + cos(x))/((1 - sin(x)^2)*(1 + cos(x)))
Potencias [src] 
                     3                        3 
     /   -I*x    I*x\         /   -I*x    I*x\  
   I*\- e     + e   /       I*\- e     + e   /  
------------------------- + --------------------
  /                    2\     /     I*x    -I*x\
  |    /   -I*x    I*x\ |     |    e      e    |
  |    \- e     + e   / |   8*|1 + ---- + -----|
8*|1 + -----------------|     \     2       2  /
  \            4        /
i(eix−e−ix)38(eix2+1+e−ix2)+i(eix−e−ix)38((eix−e−ix)24+1)\frac{i \left(e^{i x} - e^{- i x}\right)^{3}}{8 \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)} + \frac{i \left(e^{i x} - e^{- i x}\right)^{3}}{8 \left(\frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4} + 1\right)} 
i*(-exp(-i*x) + exp(i*x))^3/(8*(1 + (-exp(-i*x) + exp(i*x))^2/4)) + i*(-exp(-i*x) + exp(i*x))^3/(8*(1 + exp(i*x)/2 + exp(-i*x)/2))
Denominador común [src] 
             3         5                            
          sin (x) - sin (x) + cos(x)*sin(x) + sin(x)
-sin(x) - ------------------------------------------
                    2                  2            
            -1 + sin (x) - cos(x) + sin (x)*cos(x)
−sin⁡(x)−−sin⁡5(x)+sin⁡3(x)+sin⁡(x)cos⁡(x)+sin⁡(x)sin⁡2(x)cos⁡(x)+sin⁡2(x)−cos⁡(x)−1- \sin{\left(x \right)} - \frac{- \sin^{5}{\left(x \right)} + \sin^{3}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{\sin^{2}{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)} - \cos{\left(x \right)} - 1} 
-sin(x) - (sin(x)^3 - sin(x)^5 + cos(x)*sin(x) + sin(x))/(-1 + sin(x)^2 - cos(x) + sin(x)^2*cos(x))
Combinatoria [src] 
        3    /       2            \    
    -sin (x)*\2 - sin (x) + cos(x)/    
---------------------------------------
(1 + cos(x))*(1 + sin(x))*(-1 + sin(x))
−(−sin⁡2(x)+cos⁡(x)+2)sin⁡3(x)(sin⁡(x)−1)(sin⁡(x)+1)(cos⁡(x)+1)- \frac{\left(- \sin^{2}{\left(x \right)} + \cos{\left(x \right)} + 2\right) \sin^{3}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right) \left(\cos{\left(x \right)} + 1\right)} 
-sin(x)^3*(2 - sin(x)^2 + cos(x))/((1 + cos(x))*(1 + sin(x))*(-1 + sin(x)))
Parte trigonométrica [src] 
                 3/x\                                 3/x\             
            8*cot |-|                            8*cot |-|             
                  \2/                                  \2/             
--------------------------------- + -----------------------------------
               /            2/x\\                  /           2/x\   \
             3 |    -1 + cot |-||                3 |      4*cot |-|   |
/       2/x\\  |             \2/|   /       2/x\\  |            \2/   |
|1 + cot |-|| *|1 + ------------|   |1 + cot |-|| *|1 - --------------|
\        \2//  |           2/x\ |   \        \2//  |                 2|
               |    1 + cot |-| |                  |    /       2/x\\ |
               \            \2/ /                  |    |1 + cot |-|| |
                                                   \    \        \2// /
8cot⁡3(x2)(cot⁡2(x2)−1cot⁡2(x2)+1+1)(cot⁡2(x2)+1)3+8cot⁡3(x2)(1−4cot⁡2(x2)(cot⁡2(x2)+1)2)(cot⁡2(x2)+1)3\frac{8 \cot^{3}{\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(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} + \frac{8 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} 
                             6/x\    6/pi   x\
                       64*sin |-|*sin |-- + -|
-sin(3*x) + 3*sin(x)          \2/     \2    2/
-------------------- + -----------------------
  /       /    pi\\         3       2/    pi\ 
4*|1 + sin|x + --||      sin (x)*sin |x + --| 
  \       \    2 //                  \    2 /
3sin⁡(x)−sin⁡(3x)4(sin⁡(x+π2)+1)+64sin⁡6(x2)sin⁡6(x2+π2)sin⁡3(x)sin⁡2(x+π2)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)} + \frac{64 \sin^{6}{\left(\frac{x}{2} \right)} \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\sin^{3}{\left(x \right)} \sin^{2}{\left(x + \frac{\pi}{2} \right)}} 
         /3*x\           /x\                                           
    2*tan|---|      6*tan|-|                                           
         \ 2 /           \2/                                           
- ------------- + -----------                  2              6        
         2/3*x\          2/x\     /       2/x\\  /       2/x\\     3/x\
  1 + tan |---|   1 + tan |-|   8*|1 + tan |-|| *|1 - tan |-|| *tan |-|
          \ 2 /           \2/     \        \2//  \        \4//      \2/
----------------------------- + ---------------------------------------
       /           2/x\\                          6              2     
       |    1 - tan |-||             /       2/x\\  /       2/x\\      
       |            \2/|             |1 + tan |-|| *|1 - tan |-||      
     4*|1 + -----------|             \        \4//  \        \2//      
       |           2/x\|                                               
       |    1 + tan |-||                                               
       \            \2//
8(1−tan⁡2(x4))6(tan⁡2(x2)+1)2tan⁡3(x2)(1−tan⁡2(x2))2(tan⁡2(x4)+1)6+−2tan⁡(3x2)tan⁡2(3x2)+1+6tan⁡(x2)tan⁡2(x2)+14(1−tan⁡2(x2)tan⁡2(x2)+1+1)\frac{8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} + \frac{- \frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)} 
     3               3       
  sin (x)         sin (x)    
----------- + ---------------
       2             /    pi\
1 - sin (x)   1 + sin|x + --|
                     \    2 /
sin⁡3(x)sin⁡(x+π2)+1+sin⁡3(x)1−sin⁡2(x)\frac{\sin^{3}{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1} + \frac{\sin^{3}{\left(x \right)}}{1 - \sin^{2}{\left(x \right)}} 
         /3*x\           /x\                                      
    2*cot|---|      6*cot|-|                                      
         \ 2 /           \2/                                      
- ------------- + -----------                    2                
         2/3*x\          2/x\       /       2/x\\     3/x\    6/x\
  1 + cot |---|   1 + cot |-|   512*|1 + cot |-|| *cot |-|*cot |-|
          \ 2 /           \2/       \        \2//      \2/     \4/
----------------------------- + ----------------------------------
       /            2/x\\                      6               2  
       |    -1 + cot |-||         /       2/x\\  /        2/x\\   
       |             \2/|         |1 + cot |-|| *|-1 + cot |-||   
     4*|1 + ------------|         \        \4//  \         \2//   
       |           2/x\ |                                         
       |    1 + cot |-| |                                         
       \            \2/ /
512(cot⁡2(x2)+1)2cot⁡6(x4)cot⁡3(x2)(cot⁡2(x4)+1)6(cot⁡2(x2)−1)2+−2cot⁡(3x2)cot⁡2(3x2)+1+6cot⁡(x2)cot⁡2(x2)+14(cot⁡2(x2)−1cot⁡2(x2)+1+1)\frac{512 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cot^{6}{\left(\frac{x}{4} \right)} \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}} + \frac{- \frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)} 
     3/    pi\        3/    pi\
  cos |x - --|     cos |x - --|
      \    2 /         \    2 /
---------------- + ------------
       2/    pi\    1 + cos(x) 
1 - cos |x - --|               
        \    2 /
cos⁡3(x−π2)cos⁡(x)+1+cos⁡3(x−π2)1−cos⁡2(x−π2)\frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1} + \frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{1 - \cos^{2}{\left(x - \frac{\pi}{2} \right)}} 
     1         3              2/pi    \   
- -------- + ------      8*csc |-- - x|   
  csc(3*x)   csc(x)            \2     /   
------------------- + --------------------
  /         1     \      3/x\    3/pi   x\
4*|1 + -----------|   csc |-|*csc |-- - -|
  |       /pi    \|       \2/     \2    2/
  |    csc|-- - x||                       
  \       \2     //
8csc⁡2(−x+π2)csc⁡3(x2)csc⁡3(−x2+π2)+−1csc⁡(3x)+3csc⁡(x)4(1+1csc⁡(−x+π2))\frac{8 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{3}{\left(\frac{x}{2} \right)} \csc^{3}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{- \frac{1}{\csc{\left(3 x \right)}} + \frac{3}{\csc{\left(x \right)}}}{4 \left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)} 
                                 2      
            1                 sec (x)   
------------------------- + ------------
/      1   \    3/    pi\      3/    pi\
|1 + ------|*sec |x - --|   sec |x - --|
\    sec(x)/     \    2 /       \    2 /
sec⁡2(x)sec⁡3(x−π2)+1(1+1sec⁡(x))sec⁡3(x−π2)\frac{\sec^{2}{\left(x \right)}}{\sec^{3}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}} 
   3/    pi\      3/    pi\
cos |x - --|   cos |x - --|
    \    2 /       \    2 /
------------ + ------------
 1 + cos(x)         2      
                 cos (x)
cos⁡3(x−π2)cos⁡2(x)+cos⁡3(x−π2)cos⁡(x)+1\frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}} + \frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1} 
         /3*x\           /x\                                         
    2*tan|---|      6*tan|-|                                         
         \ 2 /           \2/                                         
- ------------- + -----------                          2             
         2/3*x\          2/x\             /       2/x\\     6/x\     
  1 + tan |---|   1 + tan |-|         512*|1 + tan |-|| *tan |-|     
          \ 2 /           \2/             \        \2//      \4/     
----------------------------- + -------------------------------------
       /           2/x\\                     6              2        
       |    1 - tan |-||        /       2/x\\  /       2/x\\     3/x\
       |            \2/|        |1 + tan |-|| *|1 - tan |-|| *tan |-|
     4*|1 + -----------|        \        \4//  \        \2//      \2/
       |           2/x\|                                             
       |    1 + tan |-||                                             
       \            \2//
−2tan⁡(3x2)tan⁡2(3x2)+1+6tan⁡(x2)tan⁡2(x2)+14(1−tan⁡2(x2)tan⁡2(x2)+1+1)+512(tan⁡2(x2)+1)2tan⁡6(x4)(1−tan⁡2(x2))2(tan⁡2(x4)+1)6tan⁡3(x2)\frac{- \frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)} + \frac{512 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{6}{\left(\frac{x}{4} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \tan^{3}{\left(\frac{x}{2} \right)}} 
     3                             
  sin (x)      -sin(3*x) + 3*sin(x)
------------ + --------------------
   2/    pi\     /       /    pi\\ 
sin |x + --|   4*|1 + sin|x + --|| 
    \    2 /     \       \    2 //
3sin⁡(x)−sin⁡(3x)4(sin⁡(x+π2)+1)+sin⁡3(x)sin⁡2(x+π2)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)} + \frac{\sin^{3}{\left(x \right)}}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}} 
   3                          
sin (x)   -sin(3*x) + 3*sin(x)
------- + --------------------
   2         4*(1 + cos(x))   
cos (x)
3sin⁡(x)−sin⁡(3x)4(cos⁡(x)+1)+sin⁡3(x)cos⁡2(x)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}} 
                                       6                 
                         /        2/x\\     12/x\    3/x\
                       8*|-1 + cot |-|| *sin  |-|*tan |-|
-sin(3*x) + 3*sin(x)     \         \4//       \4/     \2/
-------------------- + ----------------------------------
   4*(1 + cos(x))                         2              
                            /        2/x\\     4/x\      
                            |-1 + cot |-|| *sin |-|      
                            \         \2//      \2/
3sin⁡(x)−sin⁡(3x)4(cos⁡(x)+1)+8(cot⁡2(x4)−1)6sin⁡12(x4)tan⁡3(x2)(cot⁡2(x2)−1)2sin⁡4(x2)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{8 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} 
-sin(3*x) + 3*sin(x)        3/x\    2       3/x\
-------------------- + 8*cos |-|*sec (x)*sin |-|
   4*(1 + cos(x))            \2/             \2/
3sin⁡(x)−sin⁡(3x)4(cos⁡(x)+1)+8sin⁡3(x2)cos⁡3(x2)sec⁡2(x)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + 8 \sin^{3}{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)} \sec^{2}{\left(x \right)} 
                3/x\                                 3/x\             
           8*tan |-|                            8*tan |-|             
                 \2/                                  \2/             
-------------------------------- + -----------------------------------
               /           2/x\\                  /           2/x\   \
             3 |    1 - tan |-||                3 |      4*tan |-|   |
/       2/x\\  |            \2/|   /       2/x\\  |            \2/   |
|1 + tan |-|| *|1 + -----------|   |1 + tan |-|| *|1 - --------------|
\        \2//  |           2/x\|   \        \2//  |                 2|
               |    1 + tan |-||                  |    /       2/x\\ |
               \            \2//                  |    |1 + tan |-|| |
                                                  \    \        \2// /
8tan⁡3(x2)(1−tan⁡2(x2)tan⁡2(x2)+1+1)(tan⁡2(x2)+1)3+8tan⁡3(x2)(1−4tan⁡2(x2)(tan⁡2(x2)+1)2)(tan⁡2(x2)+1)3\frac{8 \tan^{3}{\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(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} + \frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} 
     /      pi\        /    pi\        3/x\    3/x   pi\
- cos|3*x - --| + 3*cos|x - --|   8*cos |-|*cos |- - --|
     \      2 /        \    2 /         \2/     \2   2 /
------------------------------- + ----------------------
         4*(1 + cos(x))                     2           
                                         cos (x)
8cos⁡3(x2)cos⁡3(x2−π2)cos⁡2(x)+3cos⁡(x−π2)−cos⁡(3x−π2)4(cos⁡(x)+1)\frac{8 \cos^{3}{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 \cos{\left(x - \frac{\pi}{2} \right)} - \cos{\left(3 x - \frac{\pi}{2} \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} 
    3                       
 sin (x)        2       3   
---------- + sec (x)*sin (x)
1 + cos(x)
sin⁡3(x)sec⁡2(x)+sin⁡3(x)cos⁡(x)+1\sin^{3}{\left(x \right)} \sec^{2}{\left(x \right)} + \frac{\sin^{3}{\left(x \right)}}{\cos{\left(x \right)} + 1} 
                                      6                 
                         /       2/x\\     12/x\    3/x\
                       8*|1 - tan |-|| *cos  |-|*tan |-|
-sin(3*x) + 3*sin(x)     \        \4//       \4/     \2/
-------------------- + ---------------------------------
   4*(1 + cos(x))                         2             
                             /       2/x\\     4/x\     
                             |1 - tan |-|| *cos |-|     
                             \        \2//      \2/
8(1−tan⁡2(x4))6cos⁡12(x4)tan⁡3(x2)(1−tan⁡2(x2))2cos⁡4(x2)+3sin⁡(x)−sin⁡(3x)4(cos⁡(x)+1)\frac{8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \cos^{12}{\left(\frac{x}{4} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}} + \frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} 
         1                       1          
-------------------- + ---------------------
/      1   \    3      /       1   \    3   
|1 + ------|*csc (x)   |1 - -------|*csc (x)
\    sec(x)/           |       2   |        
                       \    csc (x)/
1(1+1sec⁡(x))csc⁡3(x)+1(1−1csc⁡2(x))csc⁡3(x)\frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc^{3}{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\csc^{2}{\left(x \right)}}\right) \csc^{3}{\left(x \right)}} 
                            3/x\    6/x\
                       8*cot |-|*sin |-|
-sin(3*x) + 3*sin(x)         \2/     \2/
-------------------- + -----------------
   4*(1 + cos(x))              2        
                            cos (x)
3sin⁡(x)−sin⁡(3x)4(cos⁡(x)+1)+8sin⁡6(x2)cot⁡3(x2)cos⁡2(x)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{8 \sin^{6}{\left(\frac{x}{2} \right)} \cot^{3}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(x \right)}} 
    3           3   
 sin (x)     sin (x)
---------- + -------
1 + cos(x)      2   
             cos (x)
sin⁡3(x)cos⁡2(x)+sin⁡3(x)cos⁡(x)+1\frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{3}{\left(x \right)}}{\cos{\left(x \right)} + 1} 
              3/x\                             3/x\            
         8*tan |-|                        8*tan |-|            
               \2/                              \2/            
---------------------------- + --------------------------------
                           2                  /           2/x\\
/       2/x\\ /       2/x\\                 3 |    1 - tan |-||
|1 + tan |-||*|1 - tan |-||    /       2/x\\  |            \2/|
\        \2// \        \2//    |1 + tan |-|| *|1 + -----------|
                               \        \2//  |           2/x\|
                                              |    1 + tan |-||
                                              \            \2//
8tan⁡3(x2)(1−tan⁡2(x2)tan⁡2(x2)+1+1)(tan⁡2(x2)+1)3+8tan⁡3(x2)(1−tan⁡2(x2))2(tan⁡2(x2)+1)\frac{8 \tan^{3}{\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(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} + \frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
       3               3      
    sin (x)         sin (x)   
--------------- + ------------
       /    pi\      2/    pi\
1 + sin|x + --|   sin |x + --|
       \    2 /       \    2 /
sin⁡3(x)sin⁡2(x+π2)+sin⁡3(x)sin⁡(x+π2)+1\frac{\sin^{3}{\left(x \right)}}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}} + \frac{\sin^{3}{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1} 
-sin(3*x) + 3*sin(x)   -sin(3*x) + 3*sin(x)
-------------------- + --------------------
  2*(1 + cos(2*x))        4*(1 + cos(x))
3sin⁡(x)−sin⁡(3x)2(cos⁡(2x)+1)+3sin⁡(x)−sin⁡(3x)4(cos⁡(x)+1)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{2 \left(\cos{\left(2 x \right)} + 1\right)} + \frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} 
            1                              1               
------------------------- + -------------------------------
/      1   \    3/    pi\   /         1      \    3/    pi\
|1 + ------|*sec |x - --|   |1 - ------------|*sec |x - --|
\    sec(x)/     \    2 /   |       2/    pi\|     \    2 /
                            |    sec |x - --||             
                            \        \    2 //
1(1−1sec⁡2(x−π2))sec⁡3(x−π2)+1(1+1sec⁡(x))sec⁡3(x−π2)\frac{1}{\left(1 - \frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}} 
                            3/x\    3/x\
                       8*cos |-|*sin |-|
-sin(3*x) + 3*sin(x)         \2/     \2/
-------------------- + -----------------
   4*(1 + cos(x))              2        
                            cos (x)
3sin⁡(x)−sin⁡(3x)4(cos⁡(x)+1)+8sin⁡3(x2)cos⁡3(x2)cos⁡2(x)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{8 \sin^{3}{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(x \right)}} 
         /3*x\           /x\                                          
    2*cot|---|      6*cot|-|                                          
         \ 2 /           \2/                                          
- ------------- + -----------                     2               6   
         2/3*x\          2/x\        /       2/x\\  /        2/x\\    
  1 + cot |---|   1 + cot |-|      8*|1 + cot |-|| *|-1 + cot |-||    
          \ 2 /           \2/        \        \2//  \         \4//    
----------------------------- + --------------------------------------
       /            2/x\\                    6               2        
       |    -1 + cot |-||       /       2/x\\  /        2/x\\     3/x\
       |             \2/|       |1 + cot |-|| *|-1 + cot |-|| *cot |-|
     4*|1 + ------------|       \        \4//  \         \2//      \2/
       |           2/x\ |                                             
       |    1 + cot |-| |                                             
       \            \2/ /
8(cot⁡2(x4)−1)6(cot⁡2(x2)+1)2(cot⁡2(x4)+1)6(cot⁡2(x2)−1)2cot⁡3(x2)+−2cot⁡(3x2)cot⁡2(3x2)+1+6cot⁡(x2)cot⁡2(x2)+14(cot⁡2(x2)−1cot⁡2(x2)+1+1)\frac{8 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \cot^{3}{\left(\frac{x}{2} \right)}} + \frac{- \frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)} 
                              3/x\    6/x\    12/x\
                       512*cot |-|*cot |-|*sin  |-|
-sin(3*x) + 3*sin(x)           \2/     \4/      \4/
-------------------- + ----------------------------
   4*(1 + cos(x))                      2           
                         /        2/x\\     4/x\   
                         |-1 + cot |-|| *sin |-|   
                         \         \2//      \2/
3sin⁡(x)−sin⁡(3x)4(cos⁡(x)+1)+512sin⁡12(x4)cot⁡6(x4)cot⁡3(x2)(cot⁡2(x2)−1)2sin⁡4(x2)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{512 \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)} \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} 
            1                         1          
------------------------- + ---------------------
/         1     \    3      /       1   \    3   
|1 + -----------|*csc (x)   |1 - -------|*csc (x)
|       /pi    \|           |       2   |        
|    csc|-- - x||           \    csc (x)/        
\       \2     //
1(1+1csc⁡(−x+π2))csc⁡3(x)+1(1−1csc⁡2(x))csc⁡3(x)\frac{1}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{3}{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\csc^{2}{\left(x \right)}}\right) \csc^{3}{\left(x \right)}} 
                            6/x\    3/x\
                       8*cos |-|*tan |-|
-sin(3*x) + 3*sin(x)         \2/     \2/
-------------------- + -----------------
   4*(1 + cos(x))              2        
                            cos (x)
3sin⁡(x)−sin⁡(3x)4(cos⁡(x)+1)+8cos⁡6(x2)tan⁡3(x2)cos⁡2(x)\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{8 \cos^{6}{\left(\frac{x}{2} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(x \right)}} 
               3/x\                              3/x\            
          8*cot |-|                         8*cot |-|            
                \2/                               \2/            
----------------------------- + ---------------------------------
                            2                  /            2/x\\
/       2/x\\ /        2/x\\                 3 |    -1 + cot |-||
|1 + cot |-||*|-1 + cot |-||    /       2/x\\  |             \2/|
\        \2// \         \2//    |1 + cot |-|| *|1 + ------------|
                                \        \2//  |           2/x\ |
                                               |    1 + cot |-| |
                                               \            \2/ /
8cot⁡3(x2)(cot⁡2(x2)−1)2(cot⁡2(x2)+1)+8cot⁡3(x2)(cot⁡2(x2)−1cot⁡2(x2)+1+1)(cot⁡2(x2)+1)3\frac{8 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{8 \cot^{3}{\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(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} 
        1              3                            
- ------------- + -----------                       
     /      pi\      /    pi\                       
  sec|3*x - --|   sec|x - --|             2         
     \      2 /      \    2 /        8*sec (x)      
----------------------------- + --------------------
          /      1   \             3/x\    3/x   pi\
        4*|1 + ------|          sec |-|*sec |- - --|
          \    sec(x)/              \2/     \2   2 /
8sec⁡2(x)sec⁡3(x2)sec⁡3(x2−π2)+−1sec⁡(3x−π2)+3sec⁡(x−π2)4(1+1sec⁡(x))\frac{8 \sec^{2}{\left(x \right)}}{\sec^{3}{\left(\frac{x}{2} \right)} \sec^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{- \frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(x - \frac{\pi}{2} \right)}}}{4 \left(1 + \frac{1}{\sec{\left(x \right)}}\right)} 
                               2/pi    \
                            csc |-- - x|
            1                   \2     /
------------------------- + ------------
/         1     \    3           3      
|1 + -----------|*csc (x)     csc (x)   
|       /pi    \|                       
|    csc|-- - x||                       
\       \2     //
csc⁡2(−x+π2)csc⁡3(x)+1(1+1csc⁡(−x+π2))csc⁡3(x)\frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{3}{\left(x \right)}} + \frac{1}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{3}{\left(x \right)}} 
1/((1 + 1/csc(pi/2 - x))*csc(x)^3) + csc(pi/2 - x)^2/csc(x)^3
    © Sr Examen – Калькулятор