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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(x)^5/((x-3)*log(2))+5*atan(x)^4*log(x-3)/((1+x^2)*log(2))

¿Cómo vas a descomponer esta atan(x)^5/((x-3)*log(2))+5*atan(x)^4*log(x-3)/((1+x^2)*log(2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
       5               4              
   atan (x)      5*atan (x)*log(x - 3)
-------------- + ---------------------
(x - 3)*log(2)      /     2\          
                    \1 + x /*log(2)
log⁡(x−3)5atan⁡4(x)(x2+1)log⁡(2)+atan⁡5(x)(x−3)log⁡(2)\frac{\log{\left(x - 3 \right)} 5 \operatorname{atan}^{4}{\left(x \right)}}{\left(x^{2} + 1\right) \log{\left(2 \right)}} + \frac{\operatorname{atan}^{5}{\left(x \right)}}{\left(x - 3\right) \log{\left(2 \right)}} 
atan(x)^5/(((x - 3)*log(2))) + ((5*atan(x)^4)*log(x - 3))/(((1 + x^2)*log(2)))
Simplificación general [src] 
    4    //     2\                                 \
atan (x)*\\1 + x /*atan(x) + 5*(-3 + x)*log(-3 + x)/
----------------------------------------------------
              /     2\                              
              \1 + x /*(-3 + x)*log(2)
(5(x−3)log⁡(x−3)+(x2+1)atan⁡(x))atan⁡4(x)(x−3)(x2+1)log⁡(2)\frac{\left(5 \left(x - 3\right) \log{\left(x - 3 \right)} + \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{4}{\left(x \right)}}{\left(x - 3\right) \left(x^{2} + 1\right) \log{\left(2 \right)}} 
atan(x)^4*((1 + x^2)*atan(x) + 5*(-3 + x)*log(-3 + x))/((1 + x^2)*(-3 + x)*log(2))
Respuesta numérica [src] 
atan(x)^5/(-2.07944154167984 + 0.693147180559945*x) + 5.0*atan(x)^4*log(x - 3)/(0.693147180559945 + 0.693147180559945*x^2)
atan(x)^5/(-2.07944154167984 + 0.693147180559945*x) + 5.0*atan(x)^4*log(x - 3)/(0.693147180559945 + 0.693147180559945*x^2)
Denominador común [src] 
    5       2     5             4                          4               
atan (x) + x *atan (x) - 15*atan (x)*log(-3 + x) + 5*x*atan (x)*log(-3 + x)
---------------------------------------------------------------------------
                                       3             2                     
               -3*log(2) + x*log(2) + x *log(2) - 3*x *log(2)
x2atan⁡5(x)+5xlog⁡(x−3)atan⁡4(x)−15log⁡(x−3)atan⁡4(x)+atan⁡5(x)x3log⁡(2)−3x2log⁡(2)+xlog⁡(2)−3log⁡(2)\frac{x^{2} \operatorname{atan}^{5}{\left(x \right)} + 5 x \log{\left(x - 3 \right)} \operatorname{atan}^{4}{\left(x \right)} - 15 \log{\left(x - 3 \right)} \operatorname{atan}^{4}{\left(x \right)} + \operatorname{atan}^{5}{\left(x \right)}}{x^{3} \log{\left(2 \right)} - 3 x^{2} \log{\left(2 \right)} + x \log{\left(2 \right)} - 3 \log{\left(2 \right)}} 
(atan(x)^5 + x^2*atan(x)^5 - 15*atan(x)^4*log(-3 + x) + 5*x*atan(x)^4*log(-3 + x))/(-3*log(2) + x*log(2) + x^3*log(2) - 3*x^2*log(2))
Unión de expresiones racionales [src] 
    4    //     2\                                 \
atan (x)*\\1 + x /*atan(x) + 5*(-3 + x)*log(-3 + x)/
----------------------------------------------------
              /     2\                              
              \1 + x /*(-3 + x)*log(2)
(5(x−3)log⁡(x−3)+(x2+1)atan⁡(x))atan⁡4(x)(x−3)(x2+1)log⁡(2)\frac{\left(5 \left(x - 3\right) \log{\left(x - 3 \right)} + \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{4}{\left(x \right)}}{\left(x - 3\right) \left(x^{2} + 1\right) \log{\left(2 \right)}} 
atan(x)^4*((1 + x^2)*atan(x) + 5*(-3 + x)*log(-3 + x))/((1 + x^2)*(-3 + x)*log(2))
Combinatoria [src] 
    4    /                   2                                    \
atan (x)*\-15*log(-3 + x) + x *atan(x) + 5*x*log(-3 + x) + atan(x)/
-------------------------------------------------------------------
                      /     2\                                     
                      \1 + x /*(-3 + x)*log(2)
(x2atan⁡(x)+5xlog⁡(x−3)−15log⁡(x−3)+atan⁡(x))atan⁡4(x)(x−3)(x2+1)log⁡(2)\frac{\left(x^{2} \operatorname{atan}{\left(x \right)} + 5 x \log{\left(x - 3 \right)} - 15 \log{\left(x - 3 \right)} + \operatorname{atan}{\left(x \right)}\right) \operatorname{atan}^{4}{\left(x \right)}}{\left(x - 3\right) \left(x^{2} + 1\right) \log{\left(2 \right)}} 
atan(x)^4*(-15*log(-3 + x) + x^2*atan(x) + 5*x*log(-3 + x) + atan(x))/((1 + x^2)*(-3 + x)*log(2))
Potencias [src] 
        5               4               
    atan (x)      5*atan (x)*log(-3 + x)
--------------- + ----------------------
(-3 + x)*log(2)      /     2\           
                     \1 + x /*log(2)
5log⁡(x−3)atan⁡4(x)(x2+1)log⁡(2)+atan⁡5(x)(x−3)log⁡(2)\frac{5 \log{\left(x - 3 \right)} \operatorname{atan}^{4}{\left(x \right)}}{\left(x^{2} + 1\right) \log{\left(2 \right)}} + \frac{\operatorname{atan}^{5}{\left(x \right)}}{\left(x - 3\right) \log{\left(2 \right)}} 
atan(x)^5/((-3 + x)*log(2)) + 5*atan(x)^4*log(-3 + x)/((1 + x^2)*log(2))
Parte trigonométrica [src] 
        5               4               
    atan (x)      5*atan (x)*log(-3 + x)
--------------- + ----------------------
(-3 + x)*log(2)      /     2\           
                     \1 + x /*log(2)
5log⁡(x−3)atan⁡4(x)(x2+1)log⁡(2)+atan⁡5(x)(x−3)log⁡(2)\frac{5 \log{\left(x - 3 \right)} \operatorname{atan}^{4}{\left(x \right)}}{\left(x^{2} + 1\right) \log{\left(2 \right)}} + \frac{\operatorname{atan}^{5}{\left(x \right)}}{\left(x - 3\right) \log{\left(2 \right)}} 
atan(x)^5/((-3 + x)*log(2)) + 5*atan(x)^4*log(-3 + x)/((1 + x^2)*log(2))
Denominador racional [src] 
    5              2     5                    4                                 4                      
atan (x)*log(2) + x *atan (x)*log(2) - 15*atan (x)*log(2)*log(-3 + x) + 5*x*atan (x)*log(2)*log(-3 + x)
-------------------------------------------------------------------------------------------------------
                                       /     2\             2                                          
                                       \1 + x /*(-3 + x)*log (2)
x2log⁡(2)atan⁡5(x)+5xlog⁡(2)log⁡(x−3)atan⁡4(x)−15log⁡(2)log⁡(x−3)atan⁡4(x)+log⁡(2)atan⁡5(x)(x−3)(x2+1)log⁡(2)2\frac{x^{2} \log{\left(2 \right)} \operatorname{atan}^{5}{\left(x \right)} + 5 x \log{\left(2 \right)} \log{\left(x - 3 \right)} \operatorname{atan}^{4}{\left(x \right)} - 15 \log{\left(2 \right)} \log{\left(x - 3 \right)} \operatorname{atan}^{4}{\left(x \right)} + \log{\left(2 \right)} \operatorname{atan}^{5}{\left(x \right)}}{\left(x - 3\right) \left(x^{2} + 1\right) \log{\left(2 \right)}^{2}} 
(atan(x)^5*log(2) + x^2*atan(x)^5*log(2) - 15*atan(x)^4*log(2)*log(-3 + x) + 5*x*atan(x)^4*log(2)*log(-3 + x))/((1 + x^2)*(-3 + x)*log(2)^2)
Abrimos la expresión [src] 
       5               4              
   atan (x)      5*atan (x)*log(x - 3)
-------------- + ---------------------
(x - 3)*log(2)      /     2\          
                    \1 + x /*log(2)
5log⁡(x−3)atan⁡4(x)(x2+1)log⁡(2)+atan⁡5(x)(x−3)log⁡(2)\frac{5 \log{\left(x - 3 \right)} \operatorname{atan}^{4}{\left(x \right)}}{\left(x^{2} + 1\right) \log{\left(2 \right)}} + \frac{\operatorname{atan}^{5}{\left(x \right)}}{\left(x - 3\right) \log{\left(2 \right)}} 
atan(x)^5/((x - 3)*log(2)) + 5*atan(x)^4*log(x - 3)/((1 + x^2)*log(2))
Compilar la expresión [src] 
        5               4              
    atan (x)      5*atan (x)*log(x - 3)
--------------- + ---------------------
(-3 + x)*log(2)      /     2\          
                     \1 + x /*log(2)
5log⁡(x−3)atan⁡4(x)(x2+1)log⁡(2)+atan⁡5(x)(x−3)log⁡(2)\frac{5 \log{\left(x - 3 \right)} \operatorname{atan}^{4}{\left(x \right)}}{\left(x^{2} + 1\right) \log{\left(2 \right)}} + \frac{\operatorname{atan}^{5}{\left(x \right)}}{\left(x - 3\right) \log{\left(2 \right)}} 
atan(x)^5/((-3 + x)*log(2)) + 5*atan(x)^4*log(x - 3)/((1 + x^2)*log(2))
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