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