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Общий знаменатель (3*log(sin(x)+1))/16-(3*l ... 8-16*sin(x)^2+8*sin(x)^4)

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Примеры

Решение

Вы ввели [src]
                                                3                 
3*log(sin(x) + 1)   3*log(sin(x) - 1)      3*sin (x) - 5*sin(x)   
----------------- - ----------------- - --------------------------
        16                  16                    2           4   
                                        8 - 16*sin (x) + 8*sin (x)
$$- \frac{3}{16} \log{\left (\sin{\left (x \right )} - 1 \right )} + \frac{3}{16} \log{\left (\sin{\left (x \right )} + 1 \right )} - \frac{3 \sin^{3}{\left (x \right )} - 5 \sin{\left (x \right )}}{- 16 \sin^{2}{\left (x \right )} + 8 + 8 \sin^{4}{\left (x \right )}}$$
Степени [src]
                                                    3                
  3*log(-1 + sin(x))   3*log(1 + sin(x))     - 3*sin (x) + 5*sin(x)  
- ------------------ + ----------------- + --------------------------
          16                   16                    2           4   
                                           8 - 16*sin (x) + 8*sin (x)
$$\frac{- 3 \sin^{3}{\left (x \right )} + 5 \sin{\left (x \right )}}{8 \sin^{4}{\left (x \right )} - 16 \sin^{2}{\left (x \right )} + 8} - \frac{3}{16} \log{\left (\sin{\left (x \right )} - 1 \right )} + \frac{3}{16} \log{\left (\sin{\left (x \right )} + 1 \right )}$$
                                                              3      
  3*log(-1 + sin(x))   3*log(1 + sin(x))     -5*sin(x) + 3*sin (x)   
- ------------------ + ----------------- - --------------------------
          16                   16                    2           4   
                                           8 - 16*sin (x) + 8*sin (x)
$$- \frac{3 \sin^{3}{\left (x \right )} - 5 \sin{\left (x \right )}}{8 \sin^{4}{\left (x \right )} - 16 \sin^{2}{\left (x \right )} + 8} - \frac{3}{16} \log{\left (\sin{\left (x \right )} - 1 \right )} + \frac{3}{16} \log{\left (\sin{\left (x \right )} + 1 \right )}$$
Численный ответ [src]
0.1875*log(sin(x) + 1) - 0.1875*log(sin(x) - 1) - (3.0*sin(x)^3 - 5.0*sin(x))/(8.0 + 8.0*sin(x)^4 - 16.0*sin(x)^2)
Рациональный знаменатель [src]
       3                                                                2                           4                            4                           2                    
- 6*sin (x) - 3*log(-1 + sin(x)) + 3*log(1 + sin(x)) + 10*sin(x) - 6*sin (x)*log(1 + sin(x)) - 3*sin (x)*log(-1 + sin(x)) + 3*sin (x)*log(1 + sin(x)) + 6*sin (x)*log(-1 + sin(x))
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                      2            4                                                                              
                                                                           16 - 32*sin (x) + 16*sin (x)
$$\frac{1}{16 \sin^{4}{\left (x \right )} - 32 \sin^{2}{\left (x \right )} + 16} \left(- 3 \log{\left (\sin{\left (x \right )} - 1 \right )} \sin^{4}{\left (x \right )} + 6 \log{\left (\sin{\left (x \right )} - 1 \right )} \sin^{2}{\left (x \right )} - 3 \log{\left (\sin{\left (x \right )} - 1 \right )} + 3 \log{\left (\sin{\left (x \right )} + 1 \right )} \sin^{4}{\left (x \right )} - 6 \log{\left (\sin{\left (x \right )} + 1 \right )} \sin^{2}{\left (x \right )} + 3 \log{\left (\sin{\left (x \right )} + 1 \right )} - 6 \sin^{3}{\left (x \right )} + 10 \sin{\left (x \right )}\right)$$
Объединение рациональных выражений [src]
    /          2   \                                                  /       4           2   \
- 2*\-5 + 3*sin (x)/*sin(x) + 3*(-log(-1 + sin(x)) + log(1 + sin(x)))*\1 + sin (x) - 2*sin (x)/
-----------------------------------------------------------------------------------------------
                                     /       4           2   \                                 
                                  16*\1 + sin (x) - 2*sin (x)/
$$\frac{1}{16 \sin^{4}{\left (x \right )} - 32 \sin^{2}{\left (x \right )} + 16} \left(3 \left(- \log{\left (\sin{\left (x \right )} - 1 \right )} + \log{\left (\sin{\left (x \right )} + 1 \right )}\right) \left(\sin^{4}{\left (x \right )} - 2 \sin^{2}{\left (x \right )} + 1\right) - 2 \left(3 \sin^{2}{\left (x \right )} - 5\right) \sin{\left (x \right )}\right)$$
Общее упрощение [src]
       3                       4                                         
- 6*sin (x) + 10*sin(x) + 3*cos (x)*(-log(-1 + sin(x)) + log(1 + sin(x)))
-------------------------------------------------------------------------
                                      4                                  
                                16*cos (x)
$$\frac{1}{16 \cos^{4}{\left (x \right )}} \left(3 \left(- \log{\left (\sin{\left (x \right )} - 1 \right )} + \log{\left (\sin{\left (x \right )} + 1 \right )}\right) \cos^{4}{\left (x \right )} - 6 \sin^{3}{\left (x \right )} + 10 \sin{\left (x \right )}\right)$$
Собрать выражение [src]
                                                    3                
  3*log(-1 + sin(x))   3*log(1 + sin(x))     - 3*sin (x) + 5*sin(x)  
- ------------------ + ----------------- + --------------------------
          16                   16                    2           4   
                                           8 - 16*sin (x) + 8*sin (x)
$$\frac{- 3 \sin^{3}{\left (x \right )} + 5 \sin{\left (x \right )}}{8 \sin^{4}{\left (x \right )} - 16 \sin^{2}{\left (x \right )} + 8} - \frac{3}{16} \log{\left (\sin{\left (x \right )} - 1 \right )} + \frac{3}{16} \log{\left (\sin{\left (x \right )} + 1 \right )}$$
  3*log(-1 + sin(x))   3*log(1 + sin(x))             3*sin(3*x)                      11*sin(x)          
- ------------------ + ----------------- + ----------------------------- + -----------------------------
          16                   16          12 + 4*cos(4*x) + 16*cos(2*x)   12 + 4*cos(4*x) + 16*cos(2*x)
$$- \frac{3}{16} \log{\left (\sin{\left (x \right )} - 1 \right )} + \frac{3}{16} \log{\left (\sin{\left (x \right )} + 1 \right )} + \frac{11 \sin{\left (x \right )}}{16 \cos{\left (2 x \right )} + 4 \cos{\left (4 x \right )} + 12} + \frac{3 \sin{\left (3 x \right )}}{16 \cos{\left (2 x \right )} + 4 \cos{\left (4 x \right )} + 12}$$
Общий знаменатель [src]
                                                              3      
  3*log(-1 + sin(x))   3*log(1 + sin(x))     -5*sin(x) + 3*sin (x)   
- ------------------ + ----------------- - --------------------------
          16                   16                    2           4   
                                           8 - 16*sin (x) + 8*sin (x)
$$- \frac{3 \sin^{3}{\left (x \right )} - 5 \sin{\left (x \right )}}{8 \sin^{4}{\left (x \right )} - 16 \sin^{2}{\left (x \right )} + 8} - \frac{3}{16} \log{\left (\sin{\left (x \right )} - 1 \right )} + \frac{3}{16} \log{\left (\sin{\left (x \right )} + 1 \right )}$$
Тригонометрическая часть [src]
                                          /          2   \       
3*(-log(-1 + sin(x)) + log(1 + sin(x)))   \-5 + 3*sin (x)/*sin(x)
--------------------------------------- - -----------------------
                   16                                 4          
                                                 8*cos (x)
$$\frac{1}{16} \left(- 3 \log{\left (\sin{\left (x \right )} - 1 \right )} + 3 \log{\left (\sin{\left (x \right )} + 1 \right )}\right) - \frac{\sin{\left (x \right )}}{8 \cos^{4}{\left (x \right )}} \left(3 \sin^{2}{\left (x \right )} - 5\right)$$
Комбинаторика [src]
       3                                                                2                           4                            4                           2                    
- 6*sin (x) - 3*log(-1 + sin(x)) + 3*log(1 + sin(x)) + 10*sin(x) - 6*sin (x)*log(1 + sin(x)) - 3*sin (x)*log(-1 + sin(x)) + 3*sin (x)*log(1 + sin(x)) + 6*sin (x)*log(-1 + sin(x))
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                        2              2                                                                          
                                                                         16*(1 + sin(x)) *(-1 + sin(x))
$$\frac{1}{16 \left(\sin{\left (x \right )} - 1\right)^{2} \left(\sin{\left (x \right )} + 1\right)^{2}} \left(- 3 \log{\left (\sin{\left (x \right )} - 1 \right )} \sin^{4}{\left (x \right )} + 6 \log{\left (\sin{\left (x \right )} - 1 \right )} \sin^{2}{\left (x \right )} - 3 \log{\left (\sin{\left (x \right )} - 1 \right )} + 3 \log{\left (\sin{\left (x \right )} + 1 \right )} \sin^{4}{\left (x \right )} - 6 \log{\left (\sin{\left (x \right )} + 1 \right )} \sin^{2}{\left (x \right )} + 3 \log{\left (\sin{\left (x \right )} + 1 \right )} - 6 \sin^{3}{\left (x \right )} + 10 \sin{\left (x \right )}\right)$$
Раскрыть выражение [src]
                                                    3                
  3*log(-1 + sin(x))   3*log(1 + sin(x))     - 3*sin (x) + 5*sin(x)  
- ------------------ + ----------------- + --------------------------
          16                   16                    2           4   
                                           8 - 16*sin (x) + 8*sin (x)
$$\frac{- 3 \sin^{3}{\left (x \right )} + 5 \sin{\left (x \right )}}{8 \sin^{4}{\left (x \right )} - 16 \sin^{2}{\left (x \right )} + 8} - \frac{3}{16} \log{\left (\sin{\left (x \right )} - 1 \right )} + \frac{3}{16} \log{\left (\sin{\left (x \right )} + 1 \right )}$$