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Simplificación de expresiones/ Descomposición en fracciones simples/ cot(x)^atan(x)*(log(cot(x))/(1+x^2)+(-1-cot(x)^2)*atan(x)/cot(x))

¿Cómo vas a descomponer esta cot(x)^atan(x)*(log(cot(x))/(1+x^2)+(-1-cot(x)^2)*atan(x)/cot(x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
              /              /        2   \        \
   atan(x)    |log(cot(x))   \-1 - cot (x)/*atan(x)|
cot       (x)*|----------- + ----------------------|
              |        2             cot(x)        |
              \   1 + x                            /
$$\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \operatorname{atan}{\left(x \right)}}{\cot{\left(x \right)}} + \frac{\log{\left(\cot{\left(x \right)} \right)}}{x^{2} + 1}\right) \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}$$ 
cot(x)^atan(x)*(log(cot(x))/(1 + x^2) + ((-1 - cot(x)^2)*atan(x))/cot(x))
Simplificación general [src] 
                   /                     /     2\        \
   -1 + atan(x)    |                     \1 + x /*atan(x)|
cot            (x)*|cot(x)*log(cot(x)) - ----------------|
                   |                            2        |
                   \                         sin (x)     /
----------------------------------------------------------
                               2                          
                          1 + x
$$\frac{\left(- \frac{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)}\right) \cot^{\operatorname{atan}{\left(x \right)} - 1}{\left(x \right)}}{x^{2} + 1}$$ 
cot(x)^(-1 + atan(x))*(cot(x)*log(cot(x)) - (1 + x^2)*atan(x)/sin(x)^2)/(1 + x^2)
Combinatoria [src] 
    atan(x)    / 2              2                                    2    2                     \ 
-cot       (x)*\x *atan(x) + cot (x)*atan(x) - cot(x)*log(cot(x)) + x *cot (x)*atan(x) + atan(x)/ 
--------------------------------------------------------------------------------------------------
                                         /     2\                                                 
                                         \1 + x /*cot(x)
$$- \frac{\left(x^{2} \cot^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + x^{2} \operatorname{atan}{\left(x \right)} - \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right) \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \cot{\left(x \right)}}$$ 
-cot(x)^atan(x)*(x^2*atan(x) + cot(x)^2*atan(x) - cot(x)*log(cot(x)) + x^2*cot(x)^2*atan(x) + atan(x))/((1 + x^2)*cot(x))
Denominador común [src] 
     atan(x)               2    atan(x)                 atan(x)                                                     
  cot       (x)*atan(x) + x *cot       (x)*atan(x) - cot       (x)*cot(x)*log(cot(x))      atan(x)                  
- ----------------------------------------------------------------------------------- - cot       (x)*atan(x)*cot(x)
                                    2                                                                               
                                   x *cot(x) + cot(x)
$$- \cot{\left(x \right)} \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)} - \frac{x^{2} \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)} - \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)} \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} + \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x^{2} \cot{\left(x \right)} + \cot{\left(x \right)}}$$ 
-(cot(x)^atan(x)*atan(x) + x^2*cot(x)^atan(x)*atan(x) - cot(x)^atan(x)*cot(x)*log(cot(x)))/(x^2*cot(x) + cot(x)) - cot(x)^atan(x)*atan(x)*cot(x)
Unión de expresiones racionales [src] 
   atan(x)    /                     /     2\ /        2   \        \
cot       (x)*\cot(x)*log(cot(x)) + \1 + x /*\-1 - cot (x)/*atan(x)/
--------------------------------------------------------------------
                          /     2\                                  
                          \1 + x /*cot(x)
$$\frac{\left(\left(x^{2} + 1\right) \left(- \cot^{2}{\left(x \right)} - 1\right) \operatorname{atan}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)}\right) \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \cot{\left(x \right)}}$$ 
cot(x)^atan(x)*(cot(x)*log(cot(x)) + (1 + x^2)*(-1 - cot(x)^2)*atan(x))/((1 + x^2)*cot(x))
Denominador racional [src] 
   atan(x)    /                     /     2\ /        2   \        \
cot       (x)*\cot(x)*log(cot(x)) + \1 + x /*\-1 - cot (x)/*atan(x)/
--------------------------------------------------------------------
                          /     2\                                  
                          \1 + x /*cot(x)
$$\frac{\left(\left(x^{2} + 1\right) \left(- \cot^{2}{\left(x \right)} - 1\right) \operatorname{atan}{\left(x \right)} + \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)}\right) \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \cot{\left(x \right)}}$$ 
cot(x)^atan(x)*(cot(x)*log(cot(x)) + (1 + x^2)*(-1 - cot(x)^2)*atan(x))/((1 + x^2)*cot(x))
Abrimos la expresión [src] 
   atan(x)                     atan(x)                                          
cot       (x)*log(cot(x))   cot       (x)*atan(x)      atan(x)                  
------------------------- - --------------------- - cot       (x)*atan(x)*cot(x)
               2                    cot(x)                                      
          1 + x
$$- \cot{\left(x \right)} \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)} - \frac{\cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\cot{\left(x \right)}} + \frac{\log{\left(\cot{\left(x \right)} \right)} \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{x^{2} + 1}$$ 
cot(x)^atan(x)*log(cot(x))/(1 + x^2) - cot(x)^atan(x)*atan(x)/cot(x) - cot(x)^atan(x)*atan(x)*cot(x)
Respuesta numérica [src] 
cot(x)^atan(x)*(log(cot(x))/(1.0 + x^2) + (-1.0 - cot(x)^2)*atan(x)/cot(x))
cot(x)^atan(x)*(log(cot(x))/(1.0 + x^2) + (-1.0 - cot(x)^2)*atan(x)/cot(x))
Parte trigonométrica [src] 
                /   /  1   \                     \
        atan(x) |log|------|                     |
/  1   \        |   \tan(x)/                     |
|------|       *|----------- - 2*atan(x)*csc(2*x)|
\tan(x)/        |        2                       |
                \   1 + x                        /
$$\left(- 2 \operatorname{atan}{\left(x \right)} \csc{\left(2 x \right)} + \frac{\log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\tan{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}}$$ 
                     /   /   csc(x)  \                     \
                     |log|-----------|                     |
                     |   |   /pi    \|                     |
             atan(x) |   |csc|-- - x||                     |
/   csc(x)  \        |   \   \2     //                     |
|-----------|       *|---------------- - 2*atan(x)*csc(2*x)|
|   /pi    \|        |          2                          |
|csc|-- - x||        \     1 + x                           /
\   \2     //
$$\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(- 2 \operatorname{atan}{\left(x \right)} \csc{\left(2 x \right)} + \frac{\log{\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{x^{2} + 1}\right)$$ 
                     /                   /          2      \                    \
                     |   /   cos(x)  \   |       cos (x)   |            /    pi\|
                     |log|-----------|   |-1 - ------------|*atan(x)*cos|x - --||
                     |   |   /    pi\|   |        2/    pi\|            \    2 /|
             atan(x) |   |cos|x - --||   |     cos |x - --||                    |
/   cos(x)  \        |   \   \    2 //   \         \    2 //                    |
|-----------|       *|---------------- + ---------------------------------------|
|   /    pi\|        |          2                         cos(x)                |
|cos|x - --||        \     1 + x                                                /
\   \    2 //
$$\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(- \frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - 1\right) \cos{\left(x - \frac{\pi}{2} \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} \right)}}{x^{2} + 1}\right)$$ 
                   /   / sin(2*x)\            \
                   |log|---------|            |
           atan(x) |   |     2   |            |
/ sin(2*x)\        |   \2*sin (x)/   2*atan(x)|
|---------|       *|-------------- - ---------|
|     2   |        |         2        sin(2*x)|
\2*sin (x)/        \    1 + x                 /
$$\left(\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(- \frac{2 \operatorname{atan}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$ 
                     /                   /          2      \                    \
                     |   /   csc(x)  \   |       csc (x)   |            /pi    \|
                     |log|-----------|   |-1 - ------------|*atan(x)*csc|-- - x||
                     |   |   /pi    \|   |        2/pi    \|            \2     /|
             atan(x) |   |csc|-- - x||   |     csc |-- - x||                    |
/   csc(x)  \        |   \   \2     //   \         \2     //                    |
|-----------|       *|---------------- + ---------------------------------------|
|   /pi    \|        |          2                         csc(x)                |
|csc|-- - x||        \     1 + x                                                /
\   \2     //
$$\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(- \frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - 1\right) \operatorname{atan}{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{x^{2} + 1}\right)$$ 
                /              /        2   \               \
                |              |     csc (x)|               |
                |   /csc(x)\   |-1 - -------|*atan(x)*sec(x)|
        atan(x) |log|------|   |        2   |               |
/csc(x)\        |   \sec(x)/   \     sec (x)/               |
|------|       *|----------- + -----------------------------|
\sec(x)/        |        2                 csc(x)           |
                \   1 + x                                   /
$$\left(\frac{\csc{\left(x \right)}}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(- \frac{\csc^{2}{\left(x \right)}}{\sec^{2}{\left(x \right)}} - 1\right) \operatorname{atan}{\left(x \right)} \sec{\left(x \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{\csc{\left(x \right)}}{\sec{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$ 
                     /   /   /    pi\\                          \
             atan(x) |   |sec|x - --||                          |
/   /    pi\\        |   |   \    2 /|                          |
|sec|x - --||        |log|-----------|                          |
|   \    2 /|        |   \   sec(x)  /                /      pi\|
|-----------|       *|---------------- - 2*atan(x)*sec|2*x - --||
\   sec(x)  /        |          2                     \      2 /|
                     \     1 + x                                /
$$\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(- 2 \operatorname{atan}{\left(x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)} + \frac{\log{\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$ 
                /   /  1   \                                \
        atan(x) |log|------|                                |
/  1   \        |   \tan(x)/   /        1   \               |
|------|       *|----------- + |-1 - -------|*atan(x)*tan(x)|
\tan(x)/        |        2     |        2   |               |
                \   1 + x      \     tan (x)/               /
$$\left(\left(-1 - \frac{1}{\tan^{2}{\left(x \right)}}\right) \tan{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\tan{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}}$$ 
                /   /  1   \                        \
        atan(x) |log|------|   /       2   \        |
/  1   \        |   \tan(x)/   \1 + tan (x)/*atan(x)|
|------|       *|----------- - ---------------------|
\tan(x)/        |        2             tan(x)       |
                \   1 + x                           /
$$\left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)}}{\tan{\left(x \right)}} + \frac{\log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\tan{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}}$$ 
                   /                           /        2     \        \
                   |   / sin(2*x)\        2    |     sin (2*x)|        |
                   |log|---------|   2*sin (x)*|-1 - ---------|*atan(x)|
           atan(x) |   |     2   |             |          4   |        |
/ sin(2*x)\        |   \2*sin (x)/             \     4*sin (x)/        |
|---------|       *|-------------- + ----------------------------------|
|     2   |        |         2                    sin(2*x)             |
\2*sin (x)/        \    1 + x                                          /
$$\left(\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{2 \left(-1 - \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) \sin^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$ 
              /              /       2   \        \
   atan(x)    |log(cot(x))   \1 + cot (x)/*atan(x)|
cot       (x)*|----------- - ---------------------|
              |        2             cot(x)       |
              \   1 + x                           /
$$\left(- \frac{\left(\cot^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)}}{\cot{\left(x \right)}} + \frac{\log{\left(\cot{\left(x \right)} \right)}}{x^{2} + 1}\right) \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}$$ 
                /              /        2   \               \
                |              |     cos (x)|               |
                |   /cos(x)\   |-1 - -------|*atan(x)*sin(x)|
        atan(x) |log|------|   |        2   |               |
/cos(x)\        |   \sin(x)/   \     sin (x)/               |
|------|       *|----------- + -----------------------------|
\sin(x)/        |        2                 cos(x)           |
                \   1 + x                                   /
$$\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(-1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$ 
                     /   /   cos(x)  \                \
                     |log|-----------|                |
                     |   |   /    pi\|                |
             atan(x) |   |cos|x - --||                |
/   cos(x)  \        |   \   \    2 //     2*atan(x)  |
|-----------|       *|---------------- - -------------|
|   /    pi\|        |          2           /      pi\|
|cos|x - --||        |     1 + x         cos|2*x - --||
\   \    2 //        \                      \      2 //
$$\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(- \frac{2 \operatorname{atan}{\left(x \right)}}{\cos{\left(2 x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} \right)}}{x^{2} + 1}\right)$$ 
                /   /  1   \            \
        atan(x) |log|------|            |
/  1   \        |   \tan(x)/   2*atan(x)|
|------|       *|----------- - ---------|
\tan(x)/        |        2      sin(2*x)|
                \   1 + x               /
$$\left(- \frac{2 \operatorname{atan}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\tan{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}}$$ 
   atan(x)    /log(cot(x))   2*atan(x)\
cot       (x)*|----------- - ---------|
              |        2      sin(2*x)|
              \   1 + x               /
$$\left(- \frac{2 \operatorname{atan}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\cot{\left(x \right)} \right)}}{x^{2} + 1}\right) \cot^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}$$ 
                     /                   /        2/    pi\\               \
                     |   /   /    pi\\   |     sec |x - --||               |
             atan(x) |   |sec|x - --||   |         \    2 /|               |
/   /    pi\\        |   |   \    2 /|   |-1 - ------------|*atan(x)*sec(x)|
|sec|x - --||        |log|-----------|   |          2      |               |
|   \    2 /|        |   \   sec(x)  /   \       sec (x)   /               |
|-----------|       *|---------------- + ----------------------------------|
\   sec(x)  /        |          2                      /    pi\            |
                     |     1 + x                    sec|x - --|            |
                     \                                 \    2 /            /
$$\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\left(-1 - \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)} \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} \right)}}{x^{2} + 1}\right)$$ 
(sec(x - pi/2)/sec(x))^atan(x)*(log(sec(x - pi/2)/sec(x))/(1 + x^2) + (-1 - sec(x - pi/2)^2/sec(x)^2)*atan(x)*sec(x)/sec(x - pi/2))
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