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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    log(acot(x))    /  log(acot(x))       log(atan(x))  \
atan            (x)*|---------------- - ----------------|
                    |/     2\           /     2\        |
                    \\1 + x /*atan(x)   \1 + x /*acot(x)/
---------------------------------------------------------
                            2
(log⁡(acot⁡(x))(x2+1)atan⁡(x)−log⁡(atan⁡(x))(x2+1)acot⁡(x))atan⁡log⁡(acot⁡(x))(x)2\frac{\left(\frac{\log{\left(\operatorname{acot}{\left(x \right)} \right)}}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(x \right)} \right)}}{\left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}}\right) \operatorname{atan}^{\log{\left(\operatorname{acot}{\left(x \right)} \right)}}{\left(x \right)}}{2} 
(atan(x)^log(acot(x))*(log(acot(x))/(((1 + x^2)*atan(x))) - log(atan(x))/((1 + x^2)*acot(x))))/2
Simplificación general [src] 
    -1 + log(acot(x))                                                 
atan                 (x)*(acot(x)*log(acot(x)) - atan(x)*log(atan(x)))
----------------------------------------------------------------------
                            /     2\                                  
                          2*\1 + x /*acot(x)
(log⁡(acot⁡(x))acot⁡(x)−log⁡(atan⁡(x))atan⁡(x))atan⁡log⁡(acot⁡(x))−1(x)2(x2+1)acot⁡(x)\frac{\left(\log{\left(\operatorname{acot}{\left(x \right)} \right)} \operatorname{acot}{\left(x \right)} - \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{\log{\left(\operatorname{acot}{\left(x \right)} \right)} - 1}{\left(x \right)}}{2 \left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}} 
atan(x)^(-1 + log(acot(x)))*(acot(x)*log(acot(x)) - atan(x)*log(atan(x)))/(2*(1 + x^2)*acot(x))
Respuesta numérica [src] 
0.5*atan(x)^log(acot(x))*(log(acot(x))/((1.0 + x^2)*atan(x)) - log(atan(x))/((1.0 + x^2)*acot(x)))
0.5*atan(x)^log(acot(x))*(log(acot(x))/((1.0 + x^2)*atan(x)) - log(atan(x))/((1.0 + x^2)*acot(x)))
Denominador común [src] 
    log(acot(x))                               log(acot(x))                        
atan            (x)*acot(x)*log(acot(x)) - atan            (x)*atan(x)*log(atan(x))
-----------------------------------------------------------------------------------
                                             2                                     
                      2*acot(x)*atan(x) + 2*x *acot(x)*atan(x)
log⁡(acot⁡(x))acot⁡(x)atan⁡log⁡(acot⁡(x))(x)−log⁡(atan⁡(x))atan⁡(x)atan⁡log⁡(acot⁡(x))(x)2x2acot⁡(x)atan⁡(x)+2acot⁡(x)atan⁡(x)\frac{\log{\left(\operatorname{acot}{\left(x \right)} \right)} \operatorname{acot}{\left(x \right)} \operatorname{atan}^{\log{\left(\operatorname{acot}{\left(x \right)} \right)}}{\left(x \right)} - \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} \operatorname{atan}^{\log{\left(\operatorname{acot}{\left(x \right)} \right)}}{\left(x \right)}}{2 x^{2} \operatorname{acot}{\left(x \right)} \operatorname{atan}{\left(x \right)} + 2 \operatorname{acot}{\left(x \right)} \operatorname{atan}{\left(x \right)}} 
(atan(x)^log(acot(x))*acot(x)*log(acot(x)) - atan(x)^log(acot(x))*atan(x)*log(atan(x)))/(2*acot(x)*atan(x) + 2*x^2*acot(x)*atan(x))
Potencias [src] 
    log(acot(x))    /   log(acot(x))         log(atan(x))   \
atan            (x)*|------------------ - ------------------|
                    |  /     2\             /     2\        |
                    \2*\1 + x /*atan(x)   2*\1 + x /*acot(x)/
(log⁡(acot⁡(x))2(x2+1)atan⁡(x)−log⁡(atan⁡(x))2(x2+1)acot⁡(x))atan⁡log⁡(acot⁡(x))(x)\left(\frac{\log{\left(\operatorname{acot}{\left(x \right)} \right)}}{2 \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(x \right)} \right)}}{2 \left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)}}\right) \operatorname{atan}^{\log{\left(\operatorname{acot}{\left(x \right)} \right)}}{\left(x \right)} 
atan(x)^log(acot(x))*(log(acot(x))/(2*(1 + x^2)*atan(x)) - log(atan(x))/(2*(1 + x^2)*acot(x)))
Denominador racional [src] 
    log(acot(x))    //     2\                        /     2\                     \
atan            (x)*\\1 + x /*acot(x)*log(acot(x)) - \1 + x /*atan(x)*log(atan(x))/
-----------------------------------------------------------------------------------
                                      2                                            
                              /     2\                                             
                            2*\1 + x / *acot(x)*atan(x)
((x2+1)log⁡(acot⁡(x))acot⁡(x)−(x2+1)log⁡(atan⁡(x))atan⁡(x))atan⁡log⁡(acot⁡(x))(x)2(x2+1)2acot⁡(x)atan⁡(x)\frac{\left(\left(x^{2} + 1\right) \log{\left(\operatorname{acot}{\left(x \right)} \right)} \operatorname{acot}{\left(x \right)} - \left(x^{2} + 1\right) \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{\log{\left(\operatorname{acot}{\left(x \right)} \right)}}{\left(x \right)}}{2 \left(x^{2} + 1\right)^{2} \operatorname{acot}{\left(x \right)} \operatorname{atan}{\left(x \right)}} 
atan(x)^log(acot(x))*((1 + x^2)*acot(x)*log(acot(x)) - (1 + x^2)*atan(x)*log(atan(x)))/(2*(1 + x^2)^2*acot(x)*atan(x))
Combinatoria [src] 
    log(acot(x))                                                 
atan            (x)*(acot(x)*log(acot(x)) - atan(x)*log(atan(x)))
-----------------------------------------------------------------
                      /     2\                                   
                    2*\1 + x /*acot(x)*atan(x)
(log⁡(acot⁡(x))acot⁡(x)−log⁡(atan⁡(x))atan⁡(x))atan⁡log⁡(acot⁡(x))(x)2(x2+1)acot⁡(x)atan⁡(x)\frac{\left(\log{\left(\operatorname{acot}{\left(x \right)} \right)} \operatorname{acot}{\left(x \right)} - \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{\log{\left(\operatorname{acot}{\left(x \right)} \right)}}{\left(x \right)}}{2 \left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)} \operatorname{atan}{\left(x \right)}} 
atan(x)^log(acot(x))*(acot(x)*log(acot(x)) - atan(x)*log(atan(x)))/(2*(1 + x^2)*acot(x)*atan(x))
Unión de expresiones racionales [src] 
    log(acot(x))                                                 
atan            (x)*(acot(x)*log(acot(x)) - atan(x)*log(atan(x)))
-----------------------------------------------------------------
                      /     2\                                   
                    2*\1 + x /*acot(x)*atan(x)
(log⁡(acot⁡(x))acot⁡(x)−log⁡(atan⁡(x))atan⁡(x))atan⁡log⁡(acot⁡(x))(x)2(x2+1)acot⁡(x)atan⁡(x)\frac{\left(\log{\left(\operatorname{acot}{\left(x \right)} \right)} \operatorname{acot}{\left(x \right)} - \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{\log{\left(\operatorname{acot}{\left(x \right)} \right)}}{\left(x \right)}}{2 \left(x^{2} + 1\right) \operatorname{acot}{\left(x \right)} \operatorname{atan}{\left(x \right)}} 
atan(x)^log(acot(x))*(acot(x)*log(acot(x)) - atan(x)*log(atan(x)))/(2*(1 + x^2)*acot(x)*atan(x))
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