Общий знаменатель -1/(2*sqrt(x)*(1+cos(x)^2 ... (x)*sin(x)/(1+cos(x)^2)^2

Вы ввели

                            /      ___\              
            1             2*\5 - \/ x /*cos(x)*sin(x)
- --------------------- + ---------------------------
      ___ /       2   \                       2      
  2*\/ x *\1 + cos (x)/          /       2   \       
                                 \1 + cos (x)/       

$$\frac{2 \left(- \sqrt{x} + 5\right) \cos{\left (x \right )}}{\left(\cos^{2}{\left (x \right )} + 1\right)^{2}} \sin{\left (x \right )} - \frac{1}{2 \sqrt{x} \left(\cos^{2}{\left (x \right )} + 1\right)}$$

Степени

                          /         ___\              
            1             \10 - 2*\/ x /*cos(x)*sin(x)
- --------------------- + ----------------------------
    ___ /         2   \                       2       
  \/ x *\2 + 2*cos (x)/          /       2   \        
                                 \1 + cos (x)/        

$$\frac{\sin{\left (x \right )} \cos{\left (x \right )}}{\left(\cos^{2}{\left (x \right )} + 1\right)^{2}} \left(- 2 \sqrt{x} + 10\right) - \frac{1}{\sqrt{x} \left(2 \cos^{2}{\left (x \right )} + 2\right)}$$

                          /         ___\              
            1             \10 - 2*\/ x /*cos(x)*sin(x)
- --------------------- + ----------------------------
      ___ /       2   \                       2       
  2*\/ x *\1 + cos (x)/          /       2   \        
                                 \1 + cos (x)/        

$$\frac{\sin{\left (x \right )} \cos{\left (x \right )}}{\left(\cos^{2}{\left (x \right )} + 1\right)^{2}} \left(- 2 \sqrt{x} + 10\right) - \frac{1}{2 \sqrt{x} \left(\cos^{2}{\left (x \right )} + 1\right)}$$

Численный ответ

-0.5*x^(-0.5)/(1.0 + cos(x)^2) + 2.0*(5.0 - x^0.5)*cos(x)*sin(x)/(1.0 + cos(x)^2)^2

Рациональный знаменатель

               2                                                                                            
  /       2   \           3                                      ___    3                  ___              
- \1 + cos (x)/  - 4*x*cos (x)*sin(x) - 4*x*cos(x)*sin(x) + 20*\/ x *cos (x)*sin(x) + 20*\/ x *cos(x)*sin(x)
------------------------------------------------------------------------------------------------------------
                                                                3                                           
                                               ___ /       2   \                                            
                                           2*\/ x *\1 + cos (x)/                                            

$$\frac{1}{2 \sqrt{x} \left(\cos^{2}{\left (x \right )} + 1\right)^{3}} \left(20 \sqrt{x} \sin{\left (x \right )} \cos^{3}{\left (x \right )} + 20 \sqrt{x} \sin{\left (x \right )} \cos{\left (x \right )} - 4 x \sin{\left (x \right )} \cos^{3}{\left (x \right )} - 4 x \sin{\left (x \right )} \cos{\left (x \right )} - \left(\cos^{2}{\left (x \right )} + 1\right)^{2}\right)$$

Объединение рациональных выражений

        2          ___ /      ___\              
-1 - cos (x) + 4*\/ x *\5 - \/ x /*cos(x)*sin(x)
------------------------------------------------
                                  2             
                 ___ /       2   \              
             2*\/ x *\1 + cos (x)/              

$$\frac{1}{2 \sqrt{x} \left(\cos^{2}{\left (x \right )} + 1\right)^{2}} \left(4 \sqrt{x} \left(- \sqrt{x} + 5\right) \sin{\left (x \right )} \cos{\left (x \right )} - \cos^{2}{\left (x \right )} - 1\right)$$

Общее упрощение

 /       2          ___ /       ___\         \ 
-\1 + cos (x) + 2*\/ x *\-5 + \/ x /*sin(2*x)/ 
-----------------------------------------------
                                  2            
                 ___ /       2   \             
             2*\/ x *\1 + cos (x)/             

$$- \frac{1}{2 \sqrt{x} \left(\cos^{2}{\left (x \right )} + 1\right)^{2}} \left(2 \sqrt{x} \left(\sqrt{x} - 5\right) \sin{\left (2 x \right )} + \cos^{2}{\left (x \right )} + 1\right)$$

Собрать выражение

                                                        ___              
        40*sin(2*x)           -3 - cos(2*x)         8*\/ x *sin(2*x)     
--------------------------- + ------------- - ---------------------------
19 + 12*cos(2*x) + cos(4*x)          ___      19 + 12*cos(2*x) + cos(4*x)
                                 4*\/ x                                  

$$- \frac{8 \sqrt{x} \sin{\left (2 x \right )}}{12 \cos{\left (2 x \right )} + \cos{\left (4 x \right )} + 19} + \frac{40 \sin{\left (2 x \right )}}{12 \cos{\left (2 x \right )} + \cos{\left (4 x \right )} + 19} + \frac{1}{4 \sqrt{x}} \left(- \cos{\left (2 x \right )} - 3\right)$$

Общий знаменатель

 /       2           ___                                  \ 
-\1 + cos (x) - 20*\/ x *cos(x)*sin(x) + 4*x*cos(x)*sin(x)/ 
------------------------------------------------------------
            ___       ___    4          ___    2            
        2*\/ x  + 2*\/ x *cos (x) + 4*\/ x *cos (x)         

$$- \frac{- 20 \sqrt{x} \sin{\left (x \right )} \cos{\left (x \right )} + 4 x \sin{\left (x \right )} \cos{\left (x \right )} + \cos^{2}{\left (x \right )} + 1}{2 \sqrt{x} \cos^{4}{\left (x \right )} + 4 \sqrt{x} \cos^{2}{\left (x \right )} + 2 \sqrt{x}}$$

Тригонометрическая часть

                          /      ___\         
            1             \5 - \/ x /*sin(2*x)
- --------------------- + --------------------
      ___ /       2   \                   2   
  2*\/ x *\1 + cos (x)/     /        2   \    
                            \-2 + sin (x)/    

$$\frac{\left(- \sqrt{x} + 5\right) \sin{\left (2 x \right )}}{\left(\sin^{2}{\left (x \right )} - 2\right)^{2}} - \frac{1}{2 \sqrt{x} \left(\cos^{2}{\left (x \right )} + 1\right)}$$

Комбинаторика

 /       2           ___                                  \ 
-\1 + cos (x) - 20*\/ x *cos(x)*sin(x) + 4*x*cos(x)*sin(x)/ 
------------------------------------------------------------
                                        2                   
                       ___ /       2   \                    
                   2*\/ x *\1 + cos (x)/                    

$$- \frac{1}{2 \sqrt{x} \left(\cos^{2}{\left (x \right )} + 1\right)^{2}} \left(- 20 \sqrt{x} \sin{\left (x \right )} \cos{\left (x \right )} + 4 x \sin{\left (x \right )} \cos{\left (x \right )} + \cos^{2}{\left (x \right )} + 1\right)$$

Раскрыть выражение

         2      /         ___\              
  1 + cos (x)   \10 - 2*\/ x /*cos(x)*sin(x)
- ----------- + ----------------------------
        ___                         2       
    2*\/ x             /       2   \        
                       \1 + cos (x)/        

$$\frac{\sin{\left (x \right )} \cos{\left (x \right )}}{\left(\cos^{2}{\left (x \right )} + 1\right)^{2}} \left(- 2 \sqrt{x} + 10\right) - \frac{1}{2 \sqrt{x}} \left(\cos^{2}{\left (x \right )} + 1\right)$$