Общий знаменатель (-log(sin(x)^2-1)/2)+log(sin(x))-1/(2*sin(x)^2)

Вы ввели

    /   2       \                           
-log\sin (x) - 1/                      1    
------------------ + log(sin(x)) - ---------
        2                               2   
                                   2*sin (x)

$$\frac{1}{2} \left(-1 \log{\left (\sin^{2}{\left (x \right )} - 1 \right )}\right) + \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2 \sin^{2}{\left (x \right )}}$$

Степени

                 /        2   \              
      1       log\-1 + sin (x)/              
- --------- - ----------------- + log(sin(x))
       2              2                      
  2*sin (x)                                  

$$- \frac{1}{2} \log{\left (\sin^{2}{\left (x \right )} - 1 \right )} + \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2 \sin^{2}{\left (x \right )}}$$

Численный ответ

-0.5/sin(x)^2 - 0.5*log(sin(x)^2 - 1) + log(sin(x))

Рациональный знаменатель

          2       /        2   \        2               
-2 - 2*sin (x)*log\-1 + sin (x)/ + 4*sin (x)*log(sin(x))
--------------------------------------------------------
                            2                           
                       4*sin (x)                        

$$\frac{1}{4 \sin^{2}{\left (x \right )}} \left(- 2 \log{\left (\sin^{2}{\left (x \right )} - 1 \right )} \sin^{2}{\left (x \right )} + 4 \log{\left (\sin{\left (x \right )} \right )} \sin^{2}{\left (x \right )} - 2\right)$$

Объединение рациональных выражений

        2    /     /        2   \                \
-1 + sin (x)*\- log\-1 + sin (x)/ + 2*log(sin(x))/
--------------------------------------------------
                         2                        
                    2*sin (x)                     

$$\frac{1}{2 \sin^{2}{\left (x \right )}} \left(\left(- \log{\left (\sin^{2}{\left (x \right )} - 1 \right )} + 2 \log{\left (\sin{\left (x \right )} \right )}\right) \sin^{2}{\left (x \right )} - 1\right)$$

Общее упрощение

                 /        2   \              
      1       log\-1 + sin (x)/              
- --------- - ----------------- + log(sin(x))
       2              2                      
  2*sin (x)                                  

$$- \frac{1}{2} \log{\left (\sin^{2}{\left (x \right )} - 1 \right )} + \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2 \sin^{2}{\left (x \right )}}$$

Собрать выражение

         /  1   cos(2*x)\                         
      log|- - - --------|                         
  1      \  2      2    /   cos(2*x)              
- - - ------------------- + -------- + log(sin(x))
  4            2               4                  

$$- \frac{1}{2} \log{\left (- \frac{1}{2} \cos{\left (2 x \right )} - \frac{1}{2} \right )} + \log{\left (\sin{\left (x \right )} \right )} + \frac{1}{4} \cos{\left (2 x \right )} - \frac{1}{4}$$

     2         /        2   \              
  sin (x)   log\-1 + sin (x)/              
- ------- - ----------------- + log(sin(x))
     2              2                      

$$- \frac{1}{2} \log{\left (\sin^{2}{\left (x \right )} - 1 \right )} + \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2} \sin^{2}{\left (x \right )}$$

Комбинаторика

        2       /        2   \        2               
-1 - sin (x)*log\-1 + sin (x)/ + 2*sin (x)*log(sin(x))
------------------------------------------------------
                           2                          
                      2*sin (x)                       

$$\frac{1}{2 \sin^{2}{\left (x \right )}} \left(- \log{\left (\sin^{2}{\left (x \right )} - 1 \right )} \sin^{2}{\left (x \right )} + 2 \log{\left (\sin{\left (x \right )} \right )} \sin^{2}{\left (x \right )} - 1\right)$$

Общий знаменатель

                 /        2   \              
      1       log\-1 + sin (x)/              
- --------- - ----------------- + log(sin(x))
       2              2                      
  2*sin (x)                                  

$$- \frac{1}{2} \log{\left (\sin^{2}{\left (x \right )} - 1 \right )} + \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2 \sin^{2}{\left (x \right )}}$$

Тригонометрическая часть

                 /    2   \              
      1       log\-cos (x)/              
- --------- - ------------- + log(sin(x))
       2            2                    
  2*sin (x)                              

$$\log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2} \log{\left (- \cos^{2}{\left (x \right )} \right )} - \frac{1}{2 \sin^{2}{\left (x \right )}}$$

Раскрыть выражение

     2         /        2   \              
  sin (x)   log\-1 + sin (x)/              
- ------- - ----------------- + log(sin(x))
     2              2                      

$$- \frac{1}{2} \log{\left (\sin^{2}{\left (x \right )} - 1 \right )} + \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2} \sin^{2}{\left (x \right )}$$