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

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

Expresión a simplificar:

Solución

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