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Общий знаменатель 2*(sin(x)/(8*(1+cos(x)))- ... (x))^2+1))/(24*sin(x)^3))

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Примеры

Решение

Вы ввели [src]
  /                               /       2         \\
  |                             3 |  6*sin (x)      ||
  |                 (1 + cos(x)) *|------------- + 1||
  |                               |            2    ||
  |    sin(x)                     \(1 + cos(x))     /|
2*|-------------- - ---------------------------------|
  |8*(1 + cos(x))                     3              |
  \                             24*sin (x)           /
$$2 \left(- \left(1 + \frac{6 \sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) \left(\cos{\left (x \right )} + 1\right)^{3} \frac{1}{24 \sin^{3}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{8 \left(\cos{\left (x \right )} + 1\right)}\right)$$
Степени [src]
                             /           2     \
                           3 |      6*sin (x)  |
               (1 + cos(x)) *|1 + -------------|
                             |                2|
  2*sin(x)                   \    (1 + cos(x)) /
------------ - ---------------------------------
8 + 8*cos(x)                     3              
                           12*sin (x)
$$- \frac{\left(\cos{\left (x \right )} + 1\right)^{3}}{12 \sin^{3}{\left (x \right )}} \left(1 + \frac{6 \sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) + \frac{2 \sin{\left (x \right )}}{8 \cos{\left (x \right )} + 8}$$
                               /              2       \
                             3 |  1        sin (x)    |
               2*(1 + cos(x)) *|- -- - ---------------|
                               |  24                 2|
  2*sin(x)                     \       4*(1 + cos(x)) /
------------ + ----------------------------------------
8 + 8*cos(x)                      3                    
                               sin (x)
$$\frac{2 \left(\cos{\left (x \right )} + 1\right)^{3}}{\sin^{3}{\left (x \right )}} \left(- \frac{1}{24} - \frac{\sin^{2}{\left (x \right )}}{4 \left(\cos{\left (x \right )} + 1\right)^{2}}\right) + \frac{2 \sin{\left (x \right )}}{8 \cos{\left (x \right )} + 8}$$
Численный ответ [src]
2.0*sin(x)/(8.0 + 8.0*cos(x)) - 0.0833333333333333*(1.0 + cos(x))^3*(1.0 + 6.0*sin(x)^2/(1.0 + cos(x))^2)/sin(x)^3
Рациональный знаменатель [src]
               2    4                    3                /            2        2   \
48*(1 + cos(x)) *sin (x) - 2*(1 + cos(x)) *(8 + 8*cos(x))*\(1 + cos(x))  + 6*sin (x)/
-------------------------------------------------------------------------------------
                                      2                   3                          
                       24*(1 + cos(x)) *(8 + 8*cos(x))*sin (x)
$$\frac{1}{24 \left(\cos{\left (x \right )} + 1\right)^{2} \left(8 \cos{\left (x \right )} + 8\right) \sin^{3}{\left (x \right )}} \left(- 2 \left(\left(\cos{\left (x \right )} + 1\right)^{2} + 6 \sin^{2}{\left (x \right )}\right) \left(\cos{\left (x \right )} + 1\right)^{3} \left(8 \cos{\left (x \right )} + 8\right) + 48 \left(\cos{\left (x \right )} + 1\right)^{2} \sin^{4}{\left (x \right )}\right)$$
Объединение рациональных выражений [src]
     4                  2 /            2        2   \
3*sin (x) - (1 + cos(x)) *\(1 + cos(x))  + 6*sin (x)/
-----------------------------------------------------
                                  3                  
               12*(1 + cos(x))*sin (x)
$$\frac{1}{12 \left(\cos{\left (x \right )} + 1\right) \sin^{3}{\left (x \right )}} \left(- \left(\left(\cos{\left (x \right )} + 1\right)^{2} + 6 \sin^{2}{\left (x \right )}\right) \left(\cos{\left (x \right )} + 1\right)^{2} + 3 \sin^{4}{\left (x \right )}\right)$$
Общее упрощение [src]
                   2                                 
-5 + (1 - cos(2*x))  - 5*cos(x) + cos(2*x) + cos(3*x)
-----------------------------------------------------
                                  3                  
                6*(1 + cos(x))*sin (x)
$$\frac{1}{6 \left(\cos{\left (x \right )} + 1\right) \sin^{3}{\left (x \right )}} \left(\left(- \cos{\left (2 x \right )} + 1\right)^{2} - 5 \cos{\left (x \right )} + \cos{\left (2 x \right )} + \cos{\left (3 x \right )} - 5\right)$$
Собрать выражение [src]
                      1748*sin(x)                                          1542*sin(2*x)                                          740*sin(x)                                           224*sin(3*x)                                          52*sin(7*x)                                           52*sin(8*x)                                            46*sin(7*x)                                           36*sin(6*x)                                          10*sin(8*x)                                           10*sin(9*x)                                            248*sin(6*x)                                          252*sin(3*x)                                          252*sin(2*x)                                          300*sin(5*x)                                          552*sin(4*x)                                           720*sin(5*x)                                          756*sin(4*x)                   
- ---------------------------------------------------- - -------------------------------------------------- - -------------------------------------------------- - ---------------------------------------------------- - -------------------------------------------------- - ---------------------------------------------------- - ---------------------------------------------------- - -------------------------------------------------- - -------------------------------------------------- - ---------------------------------------------------- + ---------------------------------------------------- + -------------------------------------------------- + ---------------------------------------------------- + -------------------------------------------------- + ---------------------------------------------------- + ---------------------------------------------------- + --------------------------------------------------
  15360 + 1536*cos(3*x) + 9216*cos(2*x) + 23040*cos(x)   7680 + 768*cos(3*x) + 4608*cos(2*x) + 11520*cos(x)   7680 + 768*cos(3*x) + 4608*cos(2*x) + 11520*cos(x)   15360 + 1536*cos(3*x) + 9216*cos(2*x) + 23040*cos(x)   7680 + 768*cos(3*x) + 4608*cos(2*x) + 11520*cos(x)   15360 + 1536*cos(3*x) + 9216*cos(2*x) + 23040*cos(x)   15360 + 1536*cos(3*x) + 9216*cos(2*x) + 23040*cos(x)   7680 + 768*cos(3*x) + 4608*cos(2*x) + 11520*cos(x)   7680 + 768*cos(3*x) + 4608*cos(2*x) + 11520*cos(x)   15360 + 1536*cos(3*x) + 9216*cos(2*x) + 23040*cos(x)   15360 + 1536*cos(3*x) + 9216*cos(2*x) + 23040*cos(x)   7680 + 768*cos(3*x) + 4608*cos(2*x) + 11520*cos(x)   15360 + 1536*cos(3*x) + 9216*cos(2*x) + 23040*cos(x)   7680 + 768*cos(3*x) + 4608*cos(2*x) + 11520*cos(x)   15360 + 1536*cos(3*x) + 9216*cos(2*x) + 23040*cos(x)   15360 + 1536*cos(3*x) + 9216*cos(2*x) + 23040*cos(x)   7680 + 768*cos(3*x) + 4608*cos(2*x) + 11520*cos(x)
$$- \frac{1748 \sin{\left (x \right )}}{23040 \cos{\left (x \right )} + 9216 \cos{\left (2 x \right )} + 1536 \cos{\left (3 x \right )} + 15360} + \frac{252 \sin{\left (2 x \right )}}{23040 \cos{\left (x \right )} + 9216 \cos{\left (2 x \right )} + 1536 \cos{\left (3 x \right )} + 15360} - \frac{224 \sin{\left (3 x \right )}}{23040 \cos{\left (x \right )} + 9216 \cos{\left (2 x \right )} + 1536 \cos{\left (3 x \right )} + 15360} + \frac{552 \sin{\left (4 x \right )}}{23040 \cos{\left (x \right )} + 9216 \cos{\left (2 x \right )} + 1536 \cos{\left (3 x \right )} + 15360} + \frac{720 \sin{\left (5 x \right )}}{23040 \cos{\left (x \right )} + 9216 \cos{\left (2 x \right )} + 1536 \cos{\left (3 x \right )} + 15360} + \frac{248 \sin{\left (6 x \right )}}{23040 \cos{\left (x \right )} + 9216 \cos{\left (2 x \right )} + 1536 \cos{\left (3 x \right )} + 15360} - \frac{46 \sin{\left (7 x \right )}}{23040 \cos{\left (x \right )} + 9216 \cos{\left (2 x \right )} + 1536 \cos{\left (3 x \right )} + 15360} - \frac{52 \sin{\left (8 x \right )}}{23040 \cos{\left (x \right )} + 9216 \cos{\left (2 x \right )} + 1536 \cos{\left (3 x \right )} + 15360} - \frac{10 \sin{\left (9 x \right )}}{23040 \cos{\left (x \right )} + 9216 \cos{\left (2 x \right )} + 1536 \cos{\left (3 x \right )} + 15360} - \frac{740 \sin{\left (x \right )}}{11520 \cos{\left (x \right )} + 4608 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 7680} - \frac{1542 \sin{\left (2 x \right )}}{11520 \cos{\left (x \right )} + 4608 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 7680} + \frac{252 \sin{\left (3 x \right )}}{11520 \cos{\left (x \right )} + 4608 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 7680} + \frac{756 \sin{\left (4 x \right )}}{11520 \cos{\left (x \right )} + 4608 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 7680} + \frac{300 \sin{\left (5 x \right )}}{11520 \cos{\left (x \right )} + 4608 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 7680} - \frac{36 \sin{\left (6 x \right )}}{11520 \cos{\left (x \right )} + 4608 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 7680} - \frac{52 \sin{\left (7 x \right )}}{11520 \cos{\left (x \right )} + 4608 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 7680} - \frac{10 \sin{\left (8 x \right )}}{11520 \cos{\left (x \right )} + 4608 \cos{\left (2 x \right )} + 768 \cos{\left (3 x \right )} + 7680}$$
                  /       2         \                 
                3 |  6*sin (x)      |                 
  2*(1 + cos(x)) *|------------- + 1|                 
                  |            2    |                 
                  \(1 + cos(x))     /      2*sin(x)   
- ----------------------------------- + --------------
                     3                  8*(1 + cos(x))
               24*sin (x)
$$- 2 \left(1 + \frac{6 \sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) \left(\cos{\left (x \right )} + 1\right)^{3} \frac{1}{24 \sin^{3}{\left (x \right )}} + \frac{2 \sin{\left (x \right )}}{8 \left(\cos{\left (x \right )} + 1\right)}$$
Комбинаторика [src]
 /       4           4           3                      2           2           2       2            2          \ 
-\1 + cos (x) - 3*sin (x) + 4*cos (x) + 4*cos(x) + 6*cos (x) + 6*sin (x) + 6*cos (x)*sin (x) + 12*sin (x)*cos(x)/ 
------------------------------------------------------------------------------------------------------------------
                                                                3                                                 
                                             12*(1 + cos(x))*sin (x)
$$- \frac{1}{12 \left(\cos{\left (x \right )} + 1\right) \sin^{3}{\left (x \right )}} \left(- 3 \sin^{4}{\left (x \right )} + 6 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + 12 \sin^{2}{\left (x \right )} \cos{\left (x \right )} + 6 \sin^{2}{\left (x \right )} + \cos^{4}{\left (x \right )} + 4 \cos^{3}{\left (x \right )} + 6 \cos^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 1\right)$$
Общий знаменатель [src]
 /       4           4           3                      2           2           2       2            2          \ 
-\1 + cos (x) - 3*sin (x) + 4*cos (x) + 4*cos(x) + 6*cos (x) + 6*sin (x) + 6*cos (x)*sin (x) + 12*sin (x)*cos(x)/ 
------------------------------------------------------------------------------------------------------------------
                                                3            3                                                    
                                          12*sin (x) + 12*sin (x)*cos(x)
$$- \frac{1}{12 \sin^{3}{\left (x \right )} \cos{\left (x \right )} + 12 \sin^{3}{\left (x \right )}} \left(- 3 \sin^{4}{\left (x \right )} + 6 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + 12 \sin^{2}{\left (x \right )} \cos{\left (x \right )} + 6 \sin^{2}{\left (x \right )} + \cos^{4}{\left (x \right )} + 4 \cos^{3}{\left (x \right )} + 6 \cos^{2}{\left (x \right )} + 4 \cos{\left (x \right )} + 1\right)$$
Тригонометрическая часть [src]
           3           2                               
  4 - 5*cos (x) + 3*sin (x) + 9*cos(x)       sin(x)    
- ------------------------------------ + --------------
                     3                   4*(1 + cos(x))
               12*sin (x)
$$- \frac{1}{12 \sin^{3}{\left (x \right )}} \left(3 \sin^{2}{\left (x \right )} - 5 \cos^{3}{\left (x \right )} + 9 \cos{\left (x \right )} + 4\right) + \frac{\sin{\left (x \right )}}{4 \cos{\left (x \right )} + 4}$$
Раскрыть выражение [src]
                                     /           2     \
                           3    3    |      6*sin (x)  |
               (1 + cos(x)) *sin (x)*|1 + -------------|
                                     |                2|
  2*sin(x)                           \    (1 + cos(x)) /
------------ - -----------------------------------------
8 + 8*cos(x)                       12
$$- \frac{1}{12} \left(1 + \frac{6 \sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) \left(\cos{\left (x \right )} + 1\right)^{3} \sin^{3}{\left (x \right )} + \frac{2 \sin{\left (x \right )}}{8 \cos{\left (x \right )} + 8}$$
                               /       2         \
                             3 |  6*sin (x)      |
                 (1 + cos(x)) *|------------- + 1|
                               |            2    |
    sin(x)                     \(1 + cos(x))     /
-------------- - ---------------------------------
4*(1 + cos(x))                     3              
                             12*sin (x)
$$- \frac{\left(\cos{\left (x \right )} + 1\right)^{3}}{12 \sin^{3}{\left (x \right )}} \left(1 + \frac{6 \sin^{2}{\left (x \right )}}{\left(\cos{\left (x \right )} + 1\right)^{2}}\right) + \frac{\sin{\left (x \right )}}{4 \cos{\left (x \right )} + 4}$$