Общий знаменатель sqrt(e^(2*x))*atan(e^(2*x ... (e^(2*x))-1)/(1+exp(4*x))

Вы ввели

   ______                        /   ______    \
  /  2*x      / 2*x\      2*x    |  /  2*x     |
\/  E    *atan\E   /   2*e   *log\\/  E     - 1/
-------------------- + -------------------------
      ______                         4*x        
     /  2*x                     1 + e           
   \/  E     - 1                                

$$\frac{\sqrt{e^{2 x}} \operatorname{atan}{\left (e^{2 x} \right )}}{\sqrt{e^{2 x}} - 1} + \frac{2 e^{2 x}}{e^{4 x} + 1} \log{\left (\sqrt{e^{2 x}} - 1 \right )}$$

Степени

   ______                        /        ______\
  /  2*x      / 2*x\      2*x    |       /  2*x |
\/  e    *atan\e   /   2*e   *log\-1 + \/  e    /
-------------------- + --------------------------
           ______                    4*x         
          /  2*x                1 + e            
   -1 + \/  e                                    

$$\frac{2 e^{2 x}}{e^{4 x} + 1} \log{\left (\sqrt{e^{2 x}} - 1 \right )} + \frac{\sqrt{e^{2 x}} \operatorname{atan}{\left (e^{2 x} \right )}}{\sqrt{e^{2 x}} - 1}$$

Численный ответ

(2.71828182845905^(2.0*x))^0.5*atan(E^(2*x))/(-1.0 + (2.71828182845905^(2.0*x))^0.5) + 2.0*exp(2*x)*log(sqrt(E^(2*x)) - 1)/(1.0 + exp(4*x))

Рациональный знаменатель

   ______                                          /        ______\      ______                                                         /        ______\
  /  2*x      / 2*x\       / 2*x\  2*x      2*x    |       /  2*x |     /  2*x      / 2*x\  4*x       / 2*x\  2*x  4*x      2*x  2*x    |       /  2*x |
\/  e    *atan\e   / + atan\e   /*e    - 2*e   *log\-1 + \/  e    / + \/  e    *atan\e   /*e    + atan\e   /*e   *e    + 2*e   *e   *log\-1 + \/  e    /
--------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                 /     4*x\ /      2*x\                                                                 
                                                                 \1 + e   /*\-1 + e   /                                                                 

$$\frac{1}{\left(e^{2 x} - 1\right) \left(e^{4 x} + 1\right)} \left(\sqrt{e^{2 x}} e^{4 x} \operatorname{atan}{\left (e^{2 x} \right )} + e^{2 x} e^{4 x} \operatorname{atan}{\left (e^{2 x} \right )} - 2 e^{2 x} \log{\left (\sqrt{e^{2 x}} - 1 \right )} + 2 e^{2 x} e^{2 x} \log{\left (\sqrt{e^{2 x}} - 1 \right )} + e^{2 x} \operatorname{atan}{\left (e^{2 x} \right )} + \sqrt{e^{2 x}} \operatorname{atan}{\left (e^{2 x} \right )}\right)$$

Объединение рациональных выражений

   ______                           /        ______\         /        ______\
  /  2*x  /     4*x\     / 2*x\     |       /  2*x |  2*x    |       /  2*x |
\/  e    *\1 + e   /*atan\e   / + 2*\-1 + \/  e    /*e   *log\-1 + \/  e    /
-----------------------------------------------------------------------------
                                    /        ______\                         
                         /     4*x\ |       /  2*x |                         
                         \1 + e   /*\-1 + \/  e    /                         

$$\frac{1}{\left(\sqrt{e^{2 x}} - 1\right) \left(e^{4 x} + 1\right)} \left(2 \left(\sqrt{e^{2 x}} - 1\right) e^{2 x} \log{\left (\sqrt{e^{2 x}} - 1 \right )} + \left(e^{4 x} + 1\right) \sqrt{e^{2 x}} \operatorname{atan}{\left (e^{2 x} \right )}\right)$$

Общее упрощение

   ______                           /        ______\         /        ______\
  /  2*x  /     4*x\     / 2*x\     |       /  2*x |  2*x    |       /  2*x |
\/  e    *\1 + e   /*atan\e   / + 2*\-1 + \/  e    /*e   *log\-1 + \/  e    /
-----------------------------------------------------------------------------
                                    /        ______\                         
                         /     4*x\ |       /  2*x |                         
                         \1 + e   /*\-1 + \/  e    /                         

$$\frac{1}{\left(\sqrt{e^{2 x}} - 1\right) \left(e^{4 x} + 1\right)} \left(2 \left(\sqrt{e^{2 x}} - 1\right) e^{2 x} \log{\left (\sqrt{e^{2 x}} - 1 \right )} + \left(e^{4 x} + 1\right) \sqrt{e^{2 x}} \operatorname{atan}{\left (e^{2 x} \right )}\right)$$

Собрать выражение

                       /     /                2\                /                2\\                                 
   ______              |     |/        ______\ |      ______    |/        ______\ ||           ______                
  /  2*x      / 2*x\   |     ||       /  2*x | |     /  2*x     ||       /  2*x | ||  2*x     /  2*x      / 2*x\  4*x
\/  e    *atan\e   / + \- log\\-1 + \/  e    / / + \/  e    *log\\-1 + \/  e    / //*e    + \/  e    *atan\e   /*e   
---------------------------------------------------------------------------------------------------------------------
                                                ______   /        ______\                                            
                                               /  2*x    |       /  2*x |  4*x                                       
                                        -1 + \/  e     + \-1 + \/  e    /*e                                          

$$\frac{1}{\left(\sqrt{e^{2 x}} - 1\right) e^{4 x} + \sqrt{e^{2 x}} - 1} \left(\left(\sqrt{e^{2 x}} \log{\left (\left(\sqrt{e^{2 x}} - 1\right)^{2} \right )} - \log{\left (\left(\sqrt{e^{2 x}} - 1\right)^{2} \right )}\right) e^{2 x} + \sqrt{e^{2 x}} e^{4 x} \operatorname{atan}{\left (e^{2 x} \right )} + \sqrt{e^{2 x}} \operatorname{atan}{\left (e^{2 x} \right )}\right)$$

Общий знаменатель

                            /        ______\        ______         /        ______\                          
    / 2*x\  4*x      2*x    |       /  2*x |       /  2*x   2*x    |       /  2*x |       / 2*x\             
atan\e   /*e    - 2*e   *log\-1 + \/  e    / + 2*\/  e    *e   *log\-1 + \/  e    / + atan\e   /       / 2*x\
------------------------------------------------------------------------------------------------ + atan\e   /
                                     ______             ______                                               
                                    /  2*x     4*x     /  2*x   4*x                                          
                             -1 + \/  e     - e    + \/  e    *e                                             

$$\operatorname{atan}{\left (e^{2 x} \right )} + \frac{1}{\sqrt{e^{2 x}} e^{4 x} - e^{4 x} + \sqrt{e^{2 x}} - 1} \left(e^{4 x} \operatorname{atan}{\left (e^{2 x} \right )} + 2 \sqrt{e^{2 x}} e^{2 x} \log{\left (\sqrt{e^{2 x}} - 1 \right )} - 2 e^{2 x} \log{\left (\sqrt{e^{2 x}} - 1 \right )} + \operatorname{atan}{\left (e^{2 x} \right )}\right)$$

Комбинаторика

   ______                        /        ______\      ______                        ______         /        ______\
  /  2*x      / 2*x\      2*x    |       /  2*x |     /  2*x      / 2*x\  4*x       /  2*x   2*x    |       /  2*x |
\/  e    *atan\e   / - 2*e   *log\-1 + \/  e    / + \/  e    *atan\e   /*e    + 2*\/  e    *e   *log\-1 + \/  e    /
--------------------------------------------------------------------------------------------------------------------
                                                       /        ______\                                             
                                            /     4*x\ |       /  2*x |                                             
                                            \1 + e   /*\-1 + \/  e    /                                             

$$\frac{1}{\left(\sqrt{e^{2 x}} - 1\right) \left(e^{4 x} + 1\right)} \left(\sqrt{e^{2 x}} e^{4 x} \operatorname{atan}{\left (e^{2 x} \right )} + 2 \sqrt{e^{2 x}} e^{2 x} \log{\left (\sqrt{e^{2 x}} - 1 \right )} - 2 e^{2 x} \log{\left (\sqrt{e^{2 x}} - 1 \right )} + \sqrt{e^{2 x}} \operatorname{atan}{\left (e^{2 x} \right )}\right)$$

Раскрыть выражение

   ______                        /        ______\
  /  2*x      / 2*x\      2*x    |       /  2*x |
\/  e    *atan\e   /   2*e   *log\-1 + \/  e    /
-------------------- + --------------------------
           ______                    4*x         
          /  2*x                1 + e            
   -1 + \/  e                                    

$$\frac{2 e^{2 x}}{e^{4 x} + 1} \log{\left (\sqrt{e^{2 x}} - 1 \right )} + \frac{\sqrt{e^{2 x}} \operatorname{atan}{\left (e^{2 x} \right )}}{\sqrt{e^{2 x}} - 1}$$