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Общий знаменатель -log(((2*sin(x))/(1+cos(x ... (x))/(1+cos(x))))/sqrt(5)

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Примеры

Решение

Вы ввели [src]
    /  2*sin(x)        ___    \ 
    | ---------- - 2*\/ 5  - 4| 
    | 1 + cos(x)              | 
-log|-------------------------| 
    |         ___    2*sin(x) | 
    |-4 + 2*\/ 5  + ----------| 
    \               1 + cos(x)/ 
--------------------------------
               ___              
             \/ 5
$$\frac{1}{\sqrt{5}} \left(-1 \log{\left (\frac{- 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} - 4}{-4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}\right)$$
Степени [src]
          /         ___    2*sin(x) \ 
          |-4 - 2*\/ 5  + ----------| 
   ___    |               1 + cos(x)| 
-\/ 5 *log|-------------------------| 
          |         ___    2*sin(x) | 
          |-4 + 2*\/ 5  + ----------| 
          \               1 + cos(x)/ 
--------------------------------------
                  5
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{- 2 \sqrt{5} - 4 + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{-4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Численный ответ [src]
-0.447213595499958*log(((2*sin(x))/(1 + cos(x)) - 2*sqrt(5) - 4)/(-4 + 2*sqrt(5) + (2*sin(x))/(1 + cos(x))))
Рациональный знаменатель [src]
          /   /                    ___     ___ /                    ___       ___           2*sin(x)    2*cos(x)*sin(x)\   /        ___    2*sin(x) \              ___           4*sin(x)    4*cos(x)*sin(x)\\ 
          |-2*|-8 - 8*cos(x) + 4*\/ 5  + \/ 5 *|-4 - 4*cos(x) + 2*\/ 5  + 2*\/ 5 *cos(x) + ---------- + ---------------| + |4 - 2*\/ 5  - ----------|*sin(x) + 4*\/ 5 *cos(x) + ---------- + ---------------|| 
   ___    |   \                                \                                           1 + cos(x)      1 + cos(x)  /   \              1 + cos(x)/                           1 + cos(x)      1 + cos(x)  /| 
-\/ 5 *log|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| 
          |                                              /         ___    2*sin(x) \ /                    ___       ___           2*sin(x)    2*cos(x)*sin(x)\                                               | 
          |                                              |-4 + 2*\/ 5  + ----------|*|-4 - 4*cos(x) + 2*\/ 5  + 2*\/ 5 *cos(x) + ---------- + ---------------|                                               | 
          \                                              \               1 + cos(x)/ \                                           1 + cos(x)      1 + cos(x)  /                                               / 
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                       5
$$- \frac{\sqrt{5}}{5} \log{\left (- \frac{2 \left(- 2 \sqrt{5} + 4 - \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) \sin{\left (x \right )} + 2 \sqrt{5} \left(- 4 \cos{\left (x \right )} + 2 \sqrt{5} \cos{\left (x \right )} - 4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) - 16 \cos{\left (x \right )} + 8 \sqrt{5} \cos{\left (x \right )} - 16 + 8 \sqrt{5} + \frac{8 \sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{8 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{\left(-4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right) \left(- 4 \cos{\left (x \right )} + 2 \sqrt{5} \cos{\left (x \right )} - 4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}\right)} \right )}$$
Объединение рациональных выражений [src]
          /                  ___                      \ 
   ___    |-2 - 2*cos(x) - \/ 5 *(1 + cos(x)) + sin(x)| 
-\/ 5 *log|-------------------------------------------| 
          |                  /       ___\             | 
          \     (1 + cos(x))*\-2 + \/ 5 / + sin(x)    / 
--------------------------------------------------------
                           5
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{- \sqrt{5} \left(\cos{\left (x \right )} + 1\right) + \sin{\left (x \right )} - 2 \cos{\left (x \right )} - 2}{\left(-2 + \sqrt{5}\right) \left(\cos{\left (x \right )} + 1\right) + \sin{\left (x \right )}} \right )}$$
Общее упрощение [src]
          /                       /      ___\\ 
   ___    |-sin(x) + (1 + cos(x))*\2 + \/ 5 /| 
-\/ 5 *log|----------------------------------| 
          |                       /      ___\| 
          \-sin(x) + (1 + cos(x))*\2 - \/ 5 // 
-----------------------------------------------
                       5
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{\left(2 + \sqrt{5}\right) \left(\cos{\left (x \right )} + 1\right) - \sin{\left (x \right )}}{\left(- \sqrt{5} + 2\right) \left(\cos{\left (x \right )} + 1\right) - \sin{\left (x \right )}} \right )}$$
Собрать выражение [src]
         /       ___   /       ___\                \
  ___    |-2 + \/ 5  + \-2 + \/ 5 /*cos(x) + sin(x)|
\/ 5 *log|-----------------------------------------|
         |       ___   /       ___\                |
         \-2 - \/ 5  + \-2 - \/ 5 /*cos(x) + sin(x)/
----------------------------------------------------
                         5
$$\frac{\sqrt{5}}{5} \log{\left (\frac{\sin{\left (x \right )} + \left(-2 + \sqrt{5}\right) \cos{\left (x \right )} - 2 + \sqrt{5}}{\sin{\left (x \right )} + \left(- \sqrt{5} - 2\right) \cos{\left (x \right )} - \sqrt{5} - 2} \right )}$$
          /  2*sin(x)        ___    \ 
          | ---------- - 2*\/ 5  - 4| 
   ___    | 1 + cos(x)              | 
-\/ 5 *log|-------------------------| 
          |         ___    2*sin(x) | 
          |-4 + 2*\/ 5  + ----------| 
          \               1 + cos(x)/ 
--------------------------------------
                  5
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{- 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} - 4}{-4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Общий знаменатель [src]
          /                                                                                                           ___         \ 
   ___    |             2                                            sin(x)                                         \/ 5          | 
-\/ 5 *log|- ----------------------- + ----------------------------------------------------------------- - -----------------------| 
          |         ___     sin(x)            ___                ___            sin(x)     cos(x)*sin(x)          ___     sin(x)  | 
          |  -2 + \/ 5  + ----------   -2 + \/ 5  - 2*cos(x) + \/ 5 *cos(x) + ---------- + -------------   -2 + \/ 5  + ----------| 
          \               1 + cos(x)                                          1 + cos(x)     1 + cos(x)                 1 + cos(x)/ 
------------------------------------------------------------------------------------------------------------------------------------
                                                                 5
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{\sin{\left (x \right )}}{- 2 \cos{\left (x \right )} + \sqrt{5} \cos{\left (x \right )} - 2 + \sqrt{5} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{\sqrt{5}}{-2 + \sqrt{5} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{2}{-2 + \sqrt{5} + \frac{\sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Тригонометрическая часть [src]
          /         ___    2*sin(x) \ 
          |-4 - 2*\/ 5  + ----------| 
   ___    |               1 + cos(x)| 
-\/ 5 *log|-------------------------| 
          |         ___    2*sin(x) | 
          |-4 + 2*\/ 5  + ----------| 
          \               1 + cos(x)/ 
--------------------------------------
                  5
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{- 2 \sqrt{5} - 4 + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{-4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Комбинаторика [src]
          /                                           ___                                                                                   \ 
   ___    |              4                        2*\/ 5                                             2*sin(x)                               | 
-\/ 5 *log|- ------------------------- - ------------------------- + -----------------------------------------------------------------------| 
          |           ___    2*sin(x)             ___    2*sin(x)                        ___       ___           2*sin(x)    2*cos(x)*sin(x)| 
          |  -4 + 2*\/ 5  + ----------   -4 + 2*\/ 5  + ----------   -4 - 4*cos(x) + 2*\/ 5  + 2*\/ 5 *cos(x) + ---------- + ---------------| 
          \                 1 + cos(x)                  1 + cos(x)                                              1 + cos(x)      1 + cos(x)  / 
----------------------------------------------------------------------------------------------------------------------------------------------
                                                                      5
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{2 \sin{\left (x \right )}}{- 4 \cos{\left (x \right )} + 2 \sqrt{5} \cos{\left (x \right )} - 4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} + 1} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{2 \sqrt{5}}{-4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} - \frac{4}{-4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
Раскрыть выражение [src]
          /         ___    2*sin(x) \ 
          |-4 - 2*\/ 5  + ----------| 
   ___    |               1 + cos(x)| 
-\/ 5 *log|-------------------------| 
          |         ___    2*sin(x) | 
          |-4 + 2*\/ 5  + ----------| 
          \               1 + cos(x)/ 
--------------------------------------
                  5
$$- \frac{\sqrt{5}}{5} \log{\left (\frac{- 2 \sqrt{5} - 4 + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}}{-4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1}} \right )}$$
  ___ /     /         ___    2*sin(x) \      /         ___    2*sin(x) \\
\/ 5 *|- log|-4 - 2*\/ 5  + ----------| + log|-4 + 2*\/ 5  + ----------||
      \     \               1 + cos(x)/      \               1 + cos(x)//
-------------------------------------------------------------------------
                                    5
$$\frac{\sqrt{5}}{5} \left(\log{\left (-4 + 2 \sqrt{5} + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} \right )} - \log{\left (- 2 \sqrt{5} - 4 + \frac{2 \sin{\left (x \right )}}{\cos{\left (x \right )} + 1} \right )}\right)$$