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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(tan(x))/(1+x^2)+(1+tan(x)^2)*atan(x)*cos(tan(x))

¿Cómo vas a descomponer esta sin(tan(x))/(1+x^2)+(1+tan(x)^2)*atan(x)*cos(tan(x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
sin(tan(x))   /       2   \                    
----------- + \1 + tan (x)/*atan(x)*cos(tan(x))
        2                                      
   1 + x
(tan⁡2(x)+1)atan⁡(x)cos⁡(tan⁡(x))+sin⁡(tan⁡(x))x2+1\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)} \cos{\left(\tan{\left(x \right)} \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
sin(tan(x))/(1 + x^2) + ((1 + tan(x)^2)*atan(x))*cos(tan(x))
Simplificación general [src] 
/     2\                                  
\1 + x /*atan(x)*cos(tan(x))              
---------------------------- + sin(tan(x))
             2                            
          cos (x)                         
------------------------------------------
                       2                  
                  1 + x
(x2+1)cos⁡(tan⁡(x))atan⁡(x)cos⁡2(x)+sin⁡(tan⁡(x))x2+1\frac{\frac{\left(x^{2} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
((1 + x^2)*atan(x)*cos(tan(x))/cos(x)^2 + sin(tan(x)))/(1 + x^2)
Respuesta numérica [src] 
sin(tan(x))/(1.0 + x^2) + (1.0 + tan(x)^2)*atan(x)*cos(tan(x))
sin(tan(x))/(1.0 + x^2) + (1.0 + tan(x)^2)*atan(x)*cos(tan(x))
Denominador común [src] 
sin(tan(x))                            2                       
----------- + atan(x)*cos(tan(x)) + tan (x)*atan(x)*cos(tan(x))
        2                                                      
   1 + x
cos⁡(tan⁡(x))tan⁡2(x)atan⁡(x)+cos⁡(tan⁡(x))atan⁡(x)+sin⁡(tan⁡(x))x2+1\cos{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
sin(tan(x))/(1 + x^2) + atan(x)*cos(tan(x)) + tan(x)^2*atan(x)*cos(tan(x))
Denominador racional [src] 
/     2\ /       2   \                                  
\1 + x /*\1 + tan (x)/*atan(x)*cos(tan(x)) + sin(tan(x))
--------------------------------------------------------
                              2                         
                         1 + x
(x2+1)(tan⁡2(x)+1)cos⁡(tan⁡(x))atan⁡(x)+sin⁡(tan⁡(x))x2+1\frac{\left(x^{2} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
((1 + x^2)*(1 + tan(x)^2)*atan(x)*cos(tan(x)) + sin(tan(x)))/(1 + x^2)
Unión de expresiones racionales [src] 
/     2\ /       2   \                                  
\1 + x /*\1 + tan (x)/*atan(x)*cos(tan(x)) + sin(tan(x))
--------------------------------------------------------
                              2                         
                         1 + x
(x2+1)(tan⁡2(x)+1)cos⁡(tan⁡(x))atan⁡(x)+sin⁡(tan⁡(x))x2+1\frac{\left(x^{2} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
((1 + x^2)*(1 + tan(x)^2)*atan(x)*cos(tan(x)) + sin(tan(x)))/(1 + x^2)
Combinatoria [src] 
                       2                          2                           2    2                                     
atan(x)*cos(tan(x)) + x *atan(x)*cos(tan(x)) + tan (x)*atan(x)*cos(tan(x)) + x *tan (x)*atan(x)*cos(tan(x)) + sin(tan(x))
-------------------------------------------------------------------------------------------------------------------------
                                                               2                                                         
                                                          1 + x
x2cos⁡(tan⁡(x))tan⁡2(x)atan⁡(x)+x2cos⁡(tan⁡(x))atan⁡(x)+sin⁡(tan⁡(x))+cos⁡(tan⁡(x))tan⁡2(x)atan⁡(x)+cos⁡(tan⁡(x))atan⁡(x)x2+1\frac{x^{2} \cos{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + x^{2} \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(\tan{\left(x \right)} \right)} + \cos{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}}{x^{2} + 1} 
(atan(x)*cos(tan(x)) + x^2*atan(x)*cos(tan(x)) + tan(x)^2*atan(x)*cos(tan(x)) + x^2*tan(x)^2*atan(x)*cos(tan(x)) + sin(tan(x)))/(1 + x^2)
Potencias [src] 
sin(tan(x))   /       2   \                    
----------- + \1 + tan (x)/*atan(x)*cos(tan(x))
        2                                      
   1 + x
(tan⁡2(x)+1)cos⁡(tan⁡(x))atan⁡(x)+sin⁡(tan⁡(x))x2+1\left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
                        /    I*x    -I*x     /   I*x    -I*x\ \             /      I*x    -I*x     /   I*x    -I*x\ \
                        | - e    + e        -\- e    + e    / |             |   - e    + e        -\- e    + e    / |
                        | --------------    ------------------|             |   --------------    ------------------|
/                    2\ |   I*x    -I*x         I*x    -I*x   |             |     I*x    -I*x         I*x    -I*x   |
|    /   I*x    -I*x\ | |  e    + e            e    + e       |             |    e    + e            e    + e       |
|    \- e    + e    / | |e                 e                  |           I*\- e               + e                  /
|1 - -----------------|*|--------------- + -------------------|*atan(x) - -------------------------------------------
|                   2 | \       2                   2         /                              /     2\                
|     / I*x    -I*x\  |                                                                    2*\1 + x /                
\     \e    + e    /  /
(−(−eix+e−ix)2(eix+e−ix)2+1)(e−eix+e−ixeix+e−ix2+e−−eix+e−ixeix+e−ix2)atan⁡(x)−i(−e−eix+e−ixeix+e−ix+e−−eix+e−ixeix+e−ix)2(x2+1)\left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1\right) \left(\frac{e^{\frac{- e^{i x} + e^{- i x}}{e^{i x} + e^{- i x}}}}{2} + \frac{e^{- \frac{- e^{i x} + e^{- i x}}{e^{i x} + e^{- i x}}}}{2}\right) \operatorname{atan}{\left(x \right)} - \frac{i \left(- e^{\frac{- e^{i x} + e^{- i x}}{e^{i x} + e^{- i x}}} + e^{- \frac{- e^{i x} + e^{- i x}}{e^{i x} + e^{- i x}}}\right)}{2 \left(x^{2} + 1\right)} 
(1 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp((-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))/2 + exp(-(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))/2)*atan(x) - i*(-exp((-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x))) + exp(-(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x))))/(2*(1 + x^2))
Compilar la expresión [src] 
sin(tan(x))   /       2   \                    
----------- + \1 + tan (x)/*atan(x)*cos(tan(x))
        2                                      
   1 + x
(tan⁡2(x)+1)cos⁡(tan⁡(x))atan⁡(x)+sin⁡(tan⁡(x))x2+1\left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
sin(tan(x))/(1 + x^2) + (1 + tan(x)^2)*atan(x)*cos(tan(x))
Abrimos la expresión [src] 
                      sin(tan(x))      2                       
atan(x)*cos(tan(x)) + ----------- + tan (x)*atan(x)*cos(tan(x))
                              2                                
                         1 + x
cos⁡(tan⁡(x))tan⁡2(x)atan⁡(x)+cos⁡(tan⁡(x))atan⁡(x)+sin⁡(tan⁡(x))x2+1\cos{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
atan(x)*cos(tan(x)) + sin(tan(x))/(1 + x^2) + tan(x)^2*atan(x)*cos(tan(x))
Parte trigonométrica [src] 
   /          /    pi\\   /          /          /    pi\\\        
   |       cos|x - --||   |          |       cos|x - --|||        
   |  pi      \    2 /|   |         2|  pi      \    2 /||        
cos|- -- + -----------|   |1 - 2*cos |- -- + -----------||*atan(x)
   \  2       cos(x)  /   \          \  2      2*cos(x) //        
----------------------- + ----------------------------------------
              2                              2                    
         1 + x                            cos (x)
(1−2cos⁡2(−π2+cos⁡(x−π2)2cos⁡(x)))atan⁡(x)cos⁡2(x)+cos⁡(−π2+cos⁡(x−π2)cos⁡(x))x2+1\frac{\left(1 - 2 \cos^{2}{\left(- \frac{\pi}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}} \right)}\right) \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\cos{\left(- \frac{\pi}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1} 
                            /       2/pi    \\        
                            |    csc |-- - x||        
                            |        \2     /|        
                            |1 + ------------|*atan(x)
                            |         2      |        
            1               \      csc (x)   /        
------------------------- + --------------------------
            /   /pi    \\           /pi   sec(x)\     
            |csc|-- - x||        csc|-- - ------|     
/     2\    |   \2     /|           \2    csc(x)/     
\1 + x /*csc|-----------|                             
            \   csc(x)  /
(1+csc⁡2(−x+π2)csc⁡2(x))atan⁡(x)csc⁡(π2−sec⁡(x)csc⁡(x))+1(x2+1)csc⁡(csc⁡(−x+π2)csc⁡(x))\frac{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)}}{\csc{\left(\frac{\pi}{2} - \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}} 
                                               /            2/   1    \  \        
                                             2 |       8*cot |--------|  |        
                                /       2/x\\  |             \4*cot(x)/  |        
                                |1 + cot |-|| *|1 - ---------------------|*atan(x)
                                \        \2//  |                        2|        
            /   1    \                         |    /       2/   1    \\ |        
       2*cot|--------|                         |    |1 + cot |--------|| |        
            \2*cot(x)/                         \    \        \4*cot(x)// /        
----------------------------- + --------------------------------------------------
/     2\ /       2/   1    \\                                  2                  
\1 + x /*|1 + cot |--------||                    /        2/x\\                   
         \        \2*cot(x)//                    |-1 + cot |-||                   
                                                 \         \2//
(1−8cot⁡2(14cot⁡(x))(cot⁡2(14cot⁡(x))+1)2)(cot⁡2(x2)+1)2atan⁡(x)(cot⁡2(x2)−1)2+2cot⁡(12cot⁡(x))(x2+1)(cot⁡2(12cot⁡(x))+1)\frac{\left(1 - \frac{8 \cot^{2}{\left(\frac{1}{4 \cot{\left(x \right)}} \right)}}{\left(\cot^{2}{\left(\frac{1}{4 \cot{\left(x \right)}} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \operatorname{atan}{\left(x \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}} + \frac{2 \cot{\left(\frac{1}{2 \cot{\left(x \right)}} \right)}}{\left(x^{2} + 1\right) \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1\right)} 
   /     2   \   /          /   2    \\        
   |2*sin (x)|   |         2|sin (x) ||        
sin|---------|   |1 - 2*sin |--------||*atan(x)
   \ sin(2*x)/   \          \sin(2*x)//        
-------------- + ------------------------------
         2                   2/    pi\         
    1 + x                 sin |x + --|         
                              \    2 /
(1−2sin⁡2(sin⁡2(x)sin⁡(2x)))atan⁡(x)sin⁡2(x+π2)+sin⁡(2sin⁡2(x)sin⁡(2x))x2+1\frac{\left(1 - 2 \sin^{2}{\left(\frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}\right) \operatorname{atan}{\left(x \right)}}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}} + \frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1} 
   /          /    pi\\              /   /    pi\\
   |       cos|x - --||              |cos|x - --||
   |  pi      \    2 /|              |   \    2 /|
cos|- -- + -----------|   atan(x)*cos|-----------|
   \  2       cos(x)  /              \   cos(x)  /
----------------------- + ------------------------
              2                      2            
         1 + x                    cos (x)
cos⁡(cos⁡(x−π2)cos⁡(x))atan⁡(x)cos⁡2(x)+cos⁡(−π2+cos⁡(x−π2)cos⁡(x))x2+1\frac{\cos{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)} \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\cos{\left(- \frac{\pi}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1} 
                                             /            2/tan(x)\  \        
                                           2 |       8*tan |------|  |        
                              /       2/x\\  |             \  4   /  |        
                              |1 + tan |-|| *|1 - -------------------|*atan(x)
                              \        \2//  |                      2|        
            /tan(x)\                         |    /       2/tan(x)\\ |        
       2*tan|------|                         |    |1 + tan |------|| |        
            \  2   /                         \    \        \  4   // /        
--------------------------- + ------------------------------------------------
/     2\ /       2/tan(x)\\                                 2                 
\1 + x /*|1 + tan |------||                    /       2/x\\                  
         \        \  2   //                    |1 - tan |-||                  
                                               \        \2//
(1−8tan⁡2(tan⁡(x)4)(tan⁡2(tan⁡(x)4)+1)2)(tan⁡2(x2)+1)2atan⁡(x)(1−tan⁡2(x2))2+2tan⁡(tan⁡(x)2)(x2+1)(tan⁡2(tan⁡(x)2)+1)\frac{\left(1 - \frac{8 \tan^{2}{\left(\frac{\tan{\left(x \right)}}{4} \right)}}{\left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{4} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \operatorname{atan}{\left(x \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}} + \frac{2 \tan{\left(\frac{\tan{\left(x \right)}}{2} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1\right)} 
                       /       2   \        
                       |    sec (x)|        
                       |1 + -------|*atan(x)
                       |       2   |        
         1             \    csc (x)/        
-------------------- + ---------------------
/     2\    /sec(x)\         /   sec(x)  \  
\1 + x /*csc|------|      sec|-----------|  
            \csc(x)/         |   /pi    \|  
                             |sec|-- - x||  
                             \   \2     //
(1+sec⁡2(x)csc⁡2(x))atan⁡(x)sec⁡(sec⁡(x)sec⁡(−x+π2))+1(x2+1)csc⁡(sec⁡(x)csc⁡(x))\frac{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)}}{\sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(- x + \frac{\pi}{2} \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}} 
                                /       1   \ /        2/   1    \\        
            /   1    \          |1 + -------|*|-1 + cot |--------||*atan(x)
       2*cot|--------|          |       2   | \         \2*cot(x)//        
            \2*cot(x)/          \    cot (x)/                              
----------------------------- + -------------------------------------------
/     2\ /       2/   1    \\                       2/   1    \            
\1 + x /*|1 + cot |--------||                1 + cot |--------|            
         \        \2*cot(x)//                        \2*cot(x)/
(1+1cot⁡2(x))(cot⁡2(12cot⁡(x))−1)atan⁡(x)cot⁡2(12cot⁡(x))+1+2cot⁡(12cot⁡(x))(x2+1)(cot⁡2(12cot⁡(x))+1)\frac{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} - 1\right) \operatorname{atan}{\left(x \right)}}{\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1} + \frac{2 \cot{\left(\frac{1}{2 \cot{\left(x \right)}} \right)}}{\left(x^{2} + 1\right) \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1\right)} 
                                      2            
               1                   sec (x)*atan(x) 
-------------------------------- + ----------------
/     2\    /  pi      sec(x)  \      /   sec(x)  \
\1 + x /*sec|- -- + -----------|   sec|-----------|
            |  2       /    pi\|      |   /    pi\|
            |       sec|x - --||      |sec|x - --||
            \          \    2 //      \   \    2 //
atan⁡(x)sec⁡2(x)sec⁡(sec⁡(x)sec⁡(x−π2))+1(x2+1)sec⁡(sec⁡(x)sec⁡(x−π2)−π2)\frac{\operatorname{atan}{\left(x \right)} \sec^{2}{\left(x \right)}}{\sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\pi}{2} \right)}} 
   /sin(x)\                                    
sin|------|   /       2   \                    
   \cos(x)/   |    sin (x)|            /sin(x)\
----------- + |1 + -------|*atan(x)*cos|------|
        2     |       2   |            \cos(x)/
   1 + x      \    cos (x)/
(sin⁡2(x)cos⁡2(x)+1)cos⁡(sin⁡(x)cos⁡(x))atan⁡(x)+sin⁡(sin⁡(x)cos⁡(x))x2+1\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1} 
sin(tan(x))      2                       
----------- + sec (x)*atan(x)*cos(tan(x))
        2                                
   1 + x
cos⁡(tan⁡(x))atan⁡(x)sec⁡2(x)+sin⁡(tan⁡(x))x2+1\cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} \sec^{2}{\left(x \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
                                   /         2      \        
                                   |      sec (x)   |        
                                   |1 + ------------|*atan(x)
                                   |       2/    pi\|        
                                   |    sec |x - --||        
               1                   \        \    2 //        
-------------------------------- + --------------------------
/     2\    /  pi      sec(x)  \           /   sec(x)  \     
\1 + x /*sec|- -- + -----------|        sec|-----------|     
            |  2       /    pi\|           |   /    pi\|     
            |       sec|x - --||           |sec|x - --||     
            \          \    2 //           \   \    2 //
(sec⁡2(x)sec⁡2(x−π2)+1)atan⁡(x)sec⁡(sec⁡(x)sec⁡(x−π2))+1(x2+1)sec⁡(sec⁡(x)sec⁡(x−π2)−π2)\frac{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \operatorname{atan}{\left(x \right)}}{\sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\pi}{2} \right)}} 
                                           2                           
            /tan(x)\          /       2/x\\  /       2/tan(x)\\        
       2*tan|------|          |1 + tan |-|| *|1 - tan |------||*atan(x)
            \  2   /          \        \2//  \        \  2   //        
--------------------------- + -----------------------------------------
/     2\ /       2/tan(x)\\                                       2    
\1 + x /*|1 + tan |------||       /       2/tan(x)\\ /       2/x\\     
         \        \  2   //       |1 + tan |------||*|1 - tan |-||     
                                  \        \  2   // \        \2//
2tan⁡(tan⁡(x)2)(x2+1)(tan⁡2(tan⁡(x)2)+1)+(1−tan⁡2(tan⁡(x)2))(tan⁡2(x2)+1)2atan⁡(x)(1−tan⁡2(x2))2(tan⁡2(tan⁡(x)2)+1)\frac{2 \tan{\left(\frac{\tan{\left(x \right)}}{2} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1\right)} + \frac{\left(1 - \tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \operatorname{atan}{\left(x \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1\right)} 
                               2/pi    \        
                            csc |-- - x|*atan(x)
            1                   \2     /        
------------------------- + --------------------
            /   /pi    \\        /pi   sec(x)\  
            |csc|-- - x||     csc|-- - ------|  
/     2\    |   \2     /|        \2    csc(x)/  
\1 + x /*csc|-----------|                       
            \   csc(x)  /
atan⁡(x)csc⁡2(−x+π2)csc⁡(π2−sec⁡(x)csc⁡(x))+1(x2+1)csc⁡(csc⁡(−x+π2)csc⁡(x))\frac{\operatorname{atan}{\left(x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(\frac{\pi}{2} - \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}} 
   /     2   \                                         
   |2*sin (x)|                                         
sin|---------|   /         4   \            /     2   \
   \ sin(2*x)/   |    4*sin (x)|            |2*sin (x)|
-------------- + |1 + ---------|*atan(x)*cos|---------|
         2       |       2     |            \ sin(2*x)/
    1 + x        \    sin (2*x)/
(4sin⁡4(x)sin⁡2(2x)+1)cos⁡(2sin⁡2(x)sin⁡(2x))atan⁡(x)+sin⁡(2sin⁡2(x)sin⁡(2x))x2+1\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \cos{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1} 
               1                      2    /                2             \        
-------------------------------- + sec (x)*|1 - --------------------------|*atan(x)
/     2\    /  pi      sec(x)  \           |       2/  pi       sec(x)   \|        
\1 + x /*sec|- -- + -----------|           |    sec |- -- + -------------||        
            |  2       /    pi\|           |        |  2         /    pi\||        
            |       sec|x - --||           |        |       2*sec|x - --|||        
            \          \    2 //           \        \            \    2 ///
(1−2sec⁡2(sec⁡(x)2sec⁡(x−π2)−π2))atan⁡(x)sec⁡2(x)+1(x2+1)sec⁡(sec⁡(x)sec⁡(x−π2)−π2)\left(1 - \frac{2}{\sec^{2}{\left(\frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\pi}{2} \right)}}\right) \operatorname{atan}{\left(x \right)} \sec^{2}{\left(x \right)} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\pi}{2} \right)}} 
            1                  2/pi    \ /            2        \        
------------------------- + csc |-- - x|*|1 - -----------------|*atan(x)
            /   /pi    \\       \2     / |        /   /pi    \\|        
            |csc|-- - x||                |        |csc|-- - x|||        
/     2\    |   \2     /|                |       2|   \2     /||        
\1 + x /*csc|-----------|                |    csc |-----------||        
            \   csc(x)  /                \        \  2*csc(x) //
(1−2csc⁡2(csc⁡(−x+π2)2csc⁡(x)))atan⁡(x)csc⁡2(−x+π2)+1(x2+1)csc⁡(csc⁡(−x+π2)csc⁡(x))\left(1 - \frac{2}{\csc^{2}{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}} \right)}}\right) \operatorname{atan}{\left(x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}} 
sin(tan(x))   2*atan(x)*cos(tan(x))
----------- + ---------------------
        2          1 + cos(2*x)    
   1 + x
2cos⁡(tan⁡(x))atan⁡(x)cos⁡(2x)+1+sin⁡(tan⁡(x))x2+1\frac{2 \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(2 x \right)} + 1} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
              /         2/tan(x)\\        
              |1 - 2*sin |------||*atan(x)
sin(tan(x))   \          \  2   //        
----------- + ----------------------------
        2                  2              
   1 + x                cos (x)
(1−2sin⁡2(tan⁡(x)2))atan⁡(x)cos⁡2(x)+sin⁡(tan⁡(x))x2+1\frac{\left(1 - 2 \sin^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)}\right) \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
                                             2                              
            /   1    \          /       2/x\\  /        2/   1    \\        
       2*cot|--------|          |1 + cot |-|| *|-1 + cot |--------||*atan(x)
            \2*cot(x)/          \        \2//  \         \2*cot(x)//        
----------------------------- + --------------------------------------------
/     2\ /       2/   1    \\                                          2    
\1 + x /*|1 + cot |--------||       /       2/   1    \\ /        2/x\\     
         \        \2*cot(x)//       |1 + cot |--------||*|-1 + cot |-||     
                                    \        \2*cot(x)// \         \2//
(cot⁡2(x2)+1)2(cot⁡2(12cot⁡(x))−1)atan⁡(x)(cot⁡2(x2)−1)2(cot⁡2(12cot⁡(x))+1)+2cot⁡(12cot⁡(x))(x2+1)(cot⁡2(12cot⁡(x))+1)\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} - 1\right) \operatorname{atan}{\left(x \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{1}{2 \cot{\left(x \right)}} \right)}}{\left(x^{2} + 1\right) \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1\right)} 
            /tan(x)\          /       2   \ /       2/tan(x)\\        
       2*tan|------|          \1 + tan (x)/*|1 - tan |------||*atan(x)
            \  2   /                        \        \  2   //        
--------------------------- + ----------------------------------------
/     2\ /       2/tan(x)\\                      2/tan(x)\            
\1 + x /*|1 + tan |------||               1 + tan |------|            
         \        \  2   //                       \  2   /
(1−tan⁡2(tan⁡(x)2))(tan⁡2(x)+1)atan⁡(x)tan⁡2(tan⁡(x)2)+1+2tan⁡(tan⁡(x)2)(x2+1)(tan⁡2(tan⁡(x)2)+1)\frac{\left(1 - \tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)}}{\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{\tan{\left(x \right)}}{2} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1\right)} 
   /          /    pi\\                                              
   |       cos|x - --||                                              
   |  pi      \    2 /|   /       2/    pi\\            /   /    pi\\
cos|- -- + -----------|   |    cos |x - --||            |cos|x - --||
   \  2       cos(x)  /   |        \    2 /|            |   \    2 /|
----------------------- + |1 + ------------|*atan(x)*cos|-----------|
              2           |         2      |            \   cos(x)  /
         1 + x            \      cos (x)   /
(1+cos⁡2(x−π2)cos⁡2(x))cos⁡(cos⁡(x−π2)cos⁡(x))atan⁡(x)+cos⁡(−π2+cos⁡(x−π2)cos⁡(x))x2+1\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)} \operatorname{atan}{\left(x \right)} + \frac{\cos{\left(- \frac{\pi}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1} 
sin(tan(x))   atan(x)*cos(tan(x))
----------- + -------------------
        2              2         
   1 + x            cos (x)
cos⁡(tan⁡(x))atan⁡(x)cos⁡2(x)+sin⁡(tan⁡(x))x2+1\frac{\cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1} 
   /     2   \              /          2   \
   |2*sin (x)|              |pi   2*sin (x)|
sin|---------|   atan(x)*sin|-- + ---------|
   \ sin(2*x)/              \2     sin(2*x)/
-------------- + ---------------------------
         2                  2/    pi\       
    1 + x                sin |x + --|       
                             \    2 /
sin⁡(2sin⁡2(x)sin⁡(2x)+π2)atan⁡(x)sin⁡2(x+π2)+sin⁡(2sin⁡2(x)sin⁡(2x))x2+1\frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\pi}{2} \right)} \operatorname{atan}{\left(x \right)}}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}} + \frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1} 
                            /       2   \        
                            |    sec (x)|        
                            |1 + -------|*atan(x)
                            |       2   |        
            1               \    csc (x)/        
------------------------- + ---------------------
            /   /pi    \\           /sec(x)\     
            |csc|-- - x||        sec|------|     
/     2\    |   \2     /|           \csc(x)/     
\1 + x /*csc|-----------|                        
            \   csc(x)  /
(1+sec⁡2(x)csc⁡2(x))atan⁡(x)sec⁡(sec⁡(x)csc⁡(x))+1(x2+1)csc⁡(csc⁡(−x+π2)csc⁡(x))\frac{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)}}{\sec{\left(\frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}} 
   /     2   \                                              
   |2*sin (x)|                                              
sin|---------|   /         4   \            /          2   \
   \ sin(2*x)/   |    4*sin (x)|            |pi   2*sin (x)|
-------------- + |1 + ---------|*atan(x)*sin|-- + ---------|
         2       |       2     |            \2     sin(2*x)/
    1 + x        \    sin (2*x)/
(4sin⁡4(x)sin⁡2(2x)+1)sin⁡(2sin⁡2(x)sin⁡(2x)+π2)atan⁡(x)+sin⁡(2sin⁡2(x)sin⁡(2x))x2+1\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\pi}{2} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1} 
sin(2*sin(x)^2/sin(2*x))/(1 + x^2) + (1 + 4*sin(x)^4/sin(2*x)^2)*atan(x)*sin(pi/2 + 2*sin(x)^2/sin(2*x))
    © Sr Examen – Калькулятор