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

Expresión a simplificar:

Solución

Ha introducido [src]
x _________ /-log(atan(x))            1         \
\/ atan(x) *|-------------- + ------------------|
            |       2           /     2\        |
            \      x          x*\1 + x /*atan(x)/
$$\left(\frac{1}{x \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \frac{\left(-1\right) \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{x^{2}}\right) \operatorname{atan}^{\frac{1}{x}}{\left(x \right)}$$
atan(x)^(1/x)*((-log(atan(x)))/x^2 + 1/((x*(1 + x^2))*atan(x)))
Simplificación general [src]
         1 - x                                    
         -----                                    
           x   /    /     2\                     \
(atan(x))     *\x - \1 + x /*atan(x)*log(atan(x))/
--------------------------------------------------
                    2 /     2\                    
                   x *\1 + x /                    
$$\frac{\left(x - \left(x^{2} + 1\right) \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{\frac{1 - x}{x}}{\left(x \right)}}{x^{2} \left(x^{2} + 1\right)}$$
atan(x)^((1 - x)/x)*(x - (1 + x^2)*atan(x)*log(atan(x)))/(x^2*(1 + x^2))
Respuesta numérica [src]
atan(x)^(1/x)*(-log(atan(x))/x^2 + 1/(x*(1.0 + x^2)*atan(x)))
atan(x)^(1/x)*(-log(atan(x))/x^2 + 1/(x*(1.0 + x^2)*atan(x)))
Denominador común [src]
 /    x _________   x _________                         2 x _________                     \ 
-\- x*\/ atan(x)  + \/ atan(x) *atan(x)*log(atan(x)) + x *\/ atan(x) *atan(x)*log(atan(x))/ 
--------------------------------------------------------------------------------------------
                                   2            4                                           
                                  x *atan(x) + x *atan(x)                                   
$$- \frac{x^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} \operatorname{atan}^{\frac{1}{x}}{\left(x \right)} - x \operatorname{atan}^{\frac{1}{x}}{\left(x \right)} + \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} \operatorname{atan}^{\frac{1}{x}}{\left(x \right)}}{x^{4} \operatorname{atan}{\left(x \right)} + x^{2} \operatorname{atan}{\left(x \right)}}$$
-(-x*atan(x)^(1/x) + atan(x)^(1/x)*atan(x)*log(atan(x)) + x^2*atan(x)^(1/x)*atan(x)*log(atan(x)))/(x^2*atan(x) + x^4*atan(x))
Parte trigonométrica [src]
x _________ /  log(atan(x))           1         \
\/ atan(x) *|- ------------ + ------------------|
            |        2          /     2\        |
            \       x         x*\1 + x /*atan(x)/
$$\left(\frac{1}{x \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(x \right)} \right)}}{x^{2}}\right) \operatorname{atan}^{\frac{1}{x}}{\left(x \right)}$$
atan(x)^(1/x)*(-log(atan(x))/x^2 + 1/(x*(1 + x^2)*atan(x)))
Denominador racional [src]
x _________ / 2     /     2\                     \
\/ atan(x) *\x  - x*\1 + x /*atan(x)*log(atan(x))/
--------------------------------------------------
                3 /     2\                        
               x *\1 + x /*atan(x)                
$$\frac{\left(x^{2} - x \left(x^{2} + 1\right) \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{\frac{1}{x}}{\left(x \right)}}{x^{3} \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}$$
atan(x)^(1/x)*(x^2 - x*(1 + x^2)*atan(x)*log(atan(x)))/(x^3*(1 + x^2)*atan(x))
Unión de expresiones racionales [src]
x _________ /    /     2\                     \
\/ atan(x) *\x - \1 + x /*atan(x)*log(atan(x))/
-----------------------------------------------
               2 /     2\                      
              x *\1 + x /*atan(x)              
$$\frac{\left(x - \left(x^{2} + 1\right) \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{\frac{1}{x}}{\left(x \right)}}{x^{2} \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}$$
atan(x)^(1/x)*(x - (1 + x^2)*atan(x)*log(atan(x)))/(x^2*(1 + x^2)*atan(x))
Combinatoria [src]
 x _________ /                             2                     \ 
-\/ atan(x) *\-x + atan(x)*log(atan(x)) + x *atan(x)*log(atan(x))/ 
-------------------------------------------------------------------
                         2 /     2\                                
                        x *\1 + x /*atan(x)                        
$$- \frac{\left(x^{2} \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} - x + \log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{\frac{1}{x}}{\left(x \right)}}{x^{2} \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}$$
-atan(x)^(1/x)*(-x + atan(x)*log(atan(x)) + x^2*atan(x)*log(atan(x)))/(x^2*(1 + x^2)*atan(x))
Potencias [src]
x _________ /  log(atan(x))           1         \
\/ atan(x) *|- ------------ + ------------------|
            |        2          /     2\        |
            \       x         x*\1 + x /*atan(x)/
$$\left(\frac{1}{x \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(x \right)} \right)}}{x^{2}}\right) \operatorname{atan}^{\frac{1}{x}}{\left(x \right)}$$
atan(x)^(1/x)*(-log(atan(x))/x^2 + 1/(x*(1 + x^2)*atan(x)))
Abrimos la expresión [src]
            //    1     \                 \
            ||----------|                 |
            ||  /     2\|                 |
x _________ |\x*\1 + x //   -log(atan(x)) |
\/ atan(x) *|------------ + --------------|
            |  atan(x)             2      |
            \                     x       /
$$\left(\frac{\frac{1}{x} \frac{1}{x^{2} + 1}}{\operatorname{atan}{\left(x \right)}} + \frac{\left(-1\right) \log{\left(\operatorname{atan}{\left(x \right)} \right)}}{x^{2}}\right) \operatorname{atan}^{\frac{1}{x}}{\left(x \right)}$$
atan(x)^(1/x)*((1/(x*(1 + x^2)))/atan(x) + (-log(atan(x)))/x^2)
Compilar la expresión [src]
x _________ /  log(atan(x))           1         \
\/ atan(x) *|- ------------ + ------------------|
            |        2          /     2\        |
            \       x         x*\1 + x /*atan(x)/
$$\left(\frac{1}{x \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - \frac{\log{\left(\operatorname{atan}{\left(x \right)} \right)}}{x^{2}}\right) \operatorname{atan}^{\frac{1}{x}}{\left(x \right)}$$
atan(x)^(1/x)*(-log(atan(x))/x^2 + 1/(x*(1 + x^2)*atan(x)))