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Общий знаменатель ((-log(cos(4*x)+1)/4)+log ... (4*x)/(2*cos(4*x)^2-2))/4

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Примеры

Решение

Вы ввели [src]
-log(cos(4*x) + 1)    log(cos(4*x) - 1)       cos(4*x)   
------------------- + ----------------- + ---------------
         4                    4                2         
                                          2*cos (4*x) - 2
---------------------------------------------------------
                            4
$$\frac{1}{4} \left(\frac{1}{4} \log{\left (\cos{\left (4 x \right )} - 1 \right )} + \frac{1}{4} \left(-1 \log{\left (\cos{\left (4 x \right )} + 1 \right )}\right) + \frac{\cos{\left (4 x \right )}}{2 \cos^{2}{\left (4 x \right )} - 2}\right)$$
Степени [src]
  log(1 + cos(4*x))   log(-1 + cos(4*x))         cos(4*x)      
- ----------------- + ------------------ + --------------------
          16                  16             /          2     \
                                           4*\-2 + 2*cos (4*x)/
$$\frac{1}{16} \log{\left (\cos{\left (4 x \right )} - 1 \right )} - \frac{1}{16} \log{\left (\cos{\left (4 x \right )} + 1 \right )} + \frac{\cos{\left (4 x \right )}}{8 \cos^{2}{\left (4 x \right )} - 8}$$
Численный ответ [src]
0.0625*log(cos(4*x) - 1) - 0.0625*log(cos(4*x) + 1) + 0.25*cos(4*x)/(-2.0 + 2.0*cos(4*x)^2)
Рациональный знаменатель [src]
                                      2                              2                                           
-log(-1 + cos(4*x)) + 2*cos(4*x) + cos (4*x)*log(-1 + cos(4*x)) - cos (4*x)*log(1 + cos(4*x)) + log(1 + cos(4*x))
-----------------------------------------------------------------------------------------------------------------
                                                            2                                                    
                                                -16 + 16*cos (4*x)
$$\frac{1}{16 \cos^{2}{\left (4 x \right )} - 16} \left(\log{\left (\cos{\left (4 x \right )} - 1 \right )} \cos^{2}{\left (4 x \right )} - \log{\left (\cos{\left (4 x \right )} - 1 \right )} - \log{\left (\cos{\left (4 x \right )} + 1 \right )} \cos^{2}{\left (4 x \right )} + \log{\left (\cos{\left (4 x \right )} + 1 \right )} + 2 \cos{\left (4 x \right )}\right)$$
Объединение рациональных выражений [src]
             /        2     \                                          
2*cos(4*x) + \-1 + cos (4*x)/*(-log(1 + cos(4*x)) + log(-1 + cos(4*x)))
-----------------------------------------------------------------------
                             /        2     \                          
                          16*\-1 + cos (4*x)/
$$\frac{1}{16 \cos^{2}{\left (4 x \right )} - 16} \left(\left(\log{\left (\cos{\left (4 x \right )} - 1 \right )} - \log{\left (\cos{\left (4 x \right )} + 1 \right )}\right) \left(\cos^{2}{\left (4 x \right )} - 1\right) + 2 \cos{\left (4 x \right )}\right)$$
Общее упрощение [src]
  log(1 + cos(4*x))   log(-1 + cos(4*x))     cos(4*x) 
- ----------------- + ------------------ - -----------
          16                  16                2     
                                           8*sin (4*x)
$$\frac{1}{16} \log{\left (\cos{\left (4 x \right )} - 1 \right )} - \frac{1}{16} \log{\left (\cos{\left (4 x \right )} + 1 \right )} - \frac{\cos{\left (4 x \right )}}{8 \sin^{2}{\left (4 x \right )}}$$
Собрать выражение [src]
-log(cos(4*x) + 1)          cos(4*x)        log(cos(4*x) - 1)
------------------- + ------------------- + -----------------
        4*4             /     2         \          4*4       
                      4*\2*cos (4*x) - 2/
$$\frac{1}{16} \log{\left (\cos{\left (4 x \right )} - 1 \right )} + \frac{1}{16} \left(-1 \log{\left (\cos{\left (4 x \right )} + 1 \right )}\right) + \frac{\cos{\left (4 x \right )}}{8 \cos^{2}{\left (4 x \right )} - 8}$$
-log(-1 + cos(4*x)) + 4*cos(4*x) + (-log(1 + cos(4*x)) + log(-1 + cos(4*x)))*cos(8*x) + log(1 + cos(4*x))
---------------------------------------------------------------------------------------------------------
                                            -16 + 16*cos(8*x)
$$\frac{1}{16 \cos{\left (8 x \right )} - 16} \left(\left(\log{\left (\cos{\left (4 x \right )} - 1 \right )} - \log{\left (\cos{\left (4 x \right )} + 1 \right )}\right) \cos{\left (8 x \right )} - \log{\left (\cos{\left (4 x \right )} - 1 \right )} + \log{\left (\cos{\left (4 x \right )} + 1 \right )} + 4 \cos{\left (4 x \right )}\right)$$
                2         /-1 + cos(4*x)\      / 1 + cos(4*x)\
2*cos(4*x) + cos (4*x)*log|-------------| + log|-------------|
                          \ 1 + cos(4*x)/      \-1 + cos(4*x)/
--------------------------------------------------------------
                                  2                           
                      -16 + 16*cos (4*x)
$$\frac{1}{16 \cos^{2}{\left (4 x \right )} - 16} \left(\log{\left (\frac{\cos{\left (4 x \right )} + 1}{\cos{\left (4 x \right )} - 1} \right )} + \log{\left (\frac{\cos{\left (4 x \right )} - 1}{\cos{\left (4 x \right )} + 1} \right )} \cos^{2}{\left (4 x \right )} + 2 \cos{\left (4 x \right )}\right)$$
Общий знаменатель [src]
  log(1 + cos(4*x))   log(-1 + cos(4*x))       cos(4*x)    
- ----------------- + ------------------ + ----------------
          16                  16                     2     
                                           -8 + 8*cos (4*x)
$$\frac{1}{16} \log{\left (\cos{\left (4 x \right )} - 1 \right )} - \frac{1}{16} \log{\left (\cos{\left (4 x \right )} + 1 \right )} + \frac{\cos{\left (4 x \right )}}{8 \cos^{2}{\left (4 x \right )} - 8}$$
Тригонометрическая часть [src]
  log(cos(4*x) + 1)   log(cos(4*x) - 1)     cos(4*x) 
- ----------------- + ----------------- - -----------
          16                  16               2     
                                          8*sin (4*x)
$$\frac{1}{16} \log{\left (\cos{\left (4 x \right )} - 1 \right )} - \frac{1}{16} \log{\left (\cos{\left (4 x \right )} + 1 \right )} - \frac{\cos{\left (4 x \right )}}{8 \sin^{2}{\left (4 x \right )}}$$
Комбинаторика [src]
 /                                     2                             2                                             \ 
-\-log(1 + cos(4*x)) - 2*cos(4*x) + cos (4*x)*log(1 + cos(4*x)) - cos (4*x)*log(-1 + cos(4*x)) + log(-1 + cos(4*x))/ 
---------------------------------------------------------------------------------------------------------------------
                                          16*(1 + cos(4*x))*(-1 + cos(4*x))
$$- \frac{1}{16 \left(\cos{\left (4 x \right )} - 1\right) \left(\cos{\left (4 x \right )} + 1\right)} \left(- \log{\left (\cos{\left (4 x \right )} - 1 \right )} \cos^{2}{\left (4 x \right )} + \log{\left (\cos{\left (4 x \right )} - 1 \right )} + \log{\left (\cos{\left (4 x \right )} + 1 \right )} \cos^{2}{\left (4 x \right )} - \log{\left (\cos{\left (4 x \right )} + 1 \right )} - 2 \cos{\left (4 x \right )}\right)$$
Раскрыть выражение [src]
  log(1 + cos(4*x))   log(-1 + cos(4*x))         cos(4*x)      
- ----------------- + ------------------ + --------------------
          16                  16             /          2     \
                                           4*\-2 + 2*cos (4*x)/
$$\frac{1}{16} \log{\left (\cos{\left (4 x \right )} - 1 \right )} - \frac{1}{16} \log{\left (\cos{\left (4 x \right )} + 1 \right )} + \frac{\cos{\left (4 x \right )}}{8 \cos^{2}{\left (4 x \right )} - 8}$$
     /       4         4           2       2   \      /        4         4           2       2   \             4         4           2       2          
  log\1 + cos (x) + sin (x) - 6*cos (x)*sin (x)/   log\-1 + cos (x) + sin (x) - 6*cos (x)*sin (x)/          cos (x) + sin (x) - 6*cos (x)*sin (x)       
- ---------------------------------------------- + ----------------------------------------------- + ---------------------------------------------------
                        16                                                16                           /                                              2\
                                                                                                       |       /   4         4           2       2   \ |
                                                                                                     4*\-2 + 2*\cos (x) + sin (x) - 6*cos (x)*sin (x)/ /
$$\frac{1}{16} \log{\left (\sin^{4}{\left (x \right )} - 6 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )} - 1 \right )} - \frac{1}{16} \log{\left (\sin^{4}{\left (x \right )} - 6 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )} + 1 \right )} + \frac{\sin^{4}{\left (x \right )} - 6 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )}}{8 \left(\sin^{4}{\left (x \right )} - 6 \sin^{2}{\left (x \right )} \cos^{2}{\left (x \right )} + \cos^{4}{\left (x \right )}\right)^{2} - 8}$$