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Simplificación de expresiones/ Descomposición en fracciones simples/ acot(5*x)^4/(x-5)-20*acot(5*x)^3*log(x-5)/(1+25*x^2)

¿Cómo vas a descomponer esta acot(5*x)^4/(x-5)-20*acot(5*x)^3*log(x-5)/(1+25*x^2) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    4               3                
acot (5*x)   20*acot (5*x)*log(x - 5)
---------- - ------------------------
  x - 5                     2        
                    1 + 25*x
$$- \frac{\log{\left(x - 5 \right)} 20 \operatorname{acot}^{3}{\left(5 x \right)}}{25 x^{2} + 1} + \frac{\operatorname{acot}^{4}{\left(5 x \right)}}{x - 5}$$ 
acot(5*x)^4/(x - 5) - (20*acot(5*x)^3)*log(x - 5)/(1 + 25*x^2)
Simplificación general [src] 
    3      //        2\                                    \
acot (5*x)*\\1 + 25*x /*acot(5*x) - 20*(-5 + x)*log(-5 + x)/
------------------------------------------------------------
                    /        2\                             
                    \1 + 25*x /*(-5 + x)
$$\frac{\left(- 20 \left(x - 5\right) \log{\left(x - 5 \right)} + \left(25 x^{2} + 1\right) \operatorname{acot}{\left(5 x \right)}\right) \operatorname{acot}^{3}{\left(5 x \right)}}{\left(x - 5\right) \left(25 x^{2} + 1\right)}$$ 
acot(5*x)^3*((1 + 25*x^2)*acot(5*x) - 20*(-5 + x)*log(-5 + x))/((1 + 25*x^2)*(-5 + x))
Respuesta numérica [src] 
acot(5*x)^4/(-5.0 + x) - 20.0*acot(5*x)^3*log(x - 5)/(1.0 + 25.0*x^2)
acot(5*x)^4/(-5.0 + x) - 20.0*acot(5*x)^3*log(x - 5)/(1.0 + 25.0*x^2)
Denominador común [src] 
    4            2     4                3                             3                 
acot (5*x) + 25*x *acot (5*x) + 100*acot (5*x)*log(-5 + x) - 20*x*acot (5*x)*log(-5 + x)
----------------------------------------------------------------------------------------
                                              2       3                                 
                                -5 + x - 125*x  + 25*x
$$\frac{25 x^{2} \operatorname{acot}^{4}{\left(5 x \right)} - 20 x \log{\left(x - 5 \right)} \operatorname{acot}^{3}{\left(5 x \right)} + 100 \log{\left(x - 5 \right)} \operatorname{acot}^{3}{\left(5 x \right)} + \operatorname{acot}^{4}{\left(5 x \right)}}{25 x^{3} - 125 x^{2} + x - 5}$$ 
(acot(5*x)^4 + 25*x^2*acot(5*x)^4 + 100*acot(5*x)^3*log(-5 + x) - 20*x*acot(5*x)^3*log(-5 + x))/(-5 + x - 125*x^2 + 25*x^3)
Potencias [src] 
    4               3                 
acot (5*x)   20*acot (5*x)*log(-5 + x)
---------- - -------------------------
  -5 + x                     2        
                     1 + 25*x
$$- \frac{20 \log{\left(x - 5 \right)} \operatorname{acot}^{3}{\left(5 x \right)}}{25 x^{2} + 1} + \frac{\operatorname{acot}^{4}{\left(5 x \right)}}{x - 5}$$ 
acot(5*x)^4/(-5 + x) - 20*acot(5*x)^3*log(-5 + x)/(1 + 25*x^2)
Unión de expresiones racionales [src] 
    3      //        2\                                    \
acot (5*x)*\\1 + 25*x /*acot(5*x) - 20*(-5 + x)*log(-5 + x)/
------------------------------------------------------------
                    /        2\                             
                    \1 + 25*x /*(-5 + x)
$$\frac{\left(- 20 \left(x - 5\right) \log{\left(x - 5 \right)} + \left(25 x^{2} + 1\right) \operatorname{acot}{\left(5 x \right)}\right) \operatorname{acot}^{3}{\left(5 x \right)}}{\left(x - 5\right) \left(25 x^{2} + 1\right)}$$ 
acot(5*x)^3*((1 + 25*x^2)*acot(5*x) - 20*(-5 + x)*log(-5 + x))/((1 + 25*x^2)*(-5 + x))
Parte trigonométrica [src] 
    4               3                 
acot (5*x)   20*acot (5*x)*log(-5 + x)
---------- - -------------------------
  -5 + x                     2        
                     1 + 25*x
$$- \frac{20 \log{\left(x - 5 \right)} \operatorname{acot}^{3}{\left(5 x \right)}}{25 x^{2} + 1} + \frac{\operatorname{acot}^{4}{\left(5 x \right)}}{x - 5}$$ 
acot(5*x)^4/(-5 + x) - 20*acot(5*x)^3*log(-5 + x)/(1 + 25*x^2)
Combinatoria [src] 
    3      /                                         2                      \
acot (5*x)*\100*log(-5 + x) - 20*x*log(-5 + x) + 25*x *acot(5*x) + acot(5*x)/
-----------------------------------------------------------------------------
                             /        2\                                     
                             \1 + 25*x /*(-5 + x)
$$\frac{\left(25 x^{2} \operatorname{acot}{\left(5 x \right)} - 20 x \log{\left(x - 5 \right)} + 100 \log{\left(x - 5 \right)} + \operatorname{acot}{\left(5 x \right)}\right) \operatorname{acot}^{3}{\left(5 x \right)}}{\left(x - 5\right) \left(25 x^{2} + 1\right)}$$ 
acot(5*x)^3*(100*log(-5 + x) - 20*x*log(-5 + x) + 25*x^2*acot(5*x) + acot(5*x))/((1 + 25*x^2)*(-5 + x))
Compilar la expresión [src] 
    4               3                
acot (5*x)   20*acot (5*x)*log(x - 5)
---------- - ------------------------
  -5 + x                    2        
                    1 + 25*x
$$- \frac{20 \log{\left(x - 5 \right)} \operatorname{acot}^{3}{\left(5 x \right)}}{25 x^{2} + 1} + \frac{\operatorname{acot}^{4}{\left(5 x \right)}}{x - 5}$$ 
acot(5*x)^4/(-5 + x) - 20*acot(5*x)^3*log(x - 5)/(1 + 25*x^2)
Denominador racional [src] 
    4            2     4                3                             3                 
acot (5*x) + 25*x *acot (5*x) + 100*acot (5*x)*log(-5 + x) - 20*x*acot (5*x)*log(-5 + x)
----------------------------------------------------------------------------------------
                                  /        2\                                           
                                  \1 + 25*x /*(-5 + x)
$$\frac{25 x^{2} \operatorname{acot}^{4}{\left(5 x \right)} - 20 x \log{\left(x - 5 \right)} \operatorname{acot}^{3}{\left(5 x \right)} + 100 \log{\left(x - 5 \right)} \operatorname{acot}^{3}{\left(5 x \right)} + \operatorname{acot}^{4}{\left(5 x \right)}}{\left(x - 5\right) \left(25 x^{2} + 1\right)}$$ 
(acot(5*x)^4 + 25*x^2*acot(5*x)^4 + 100*acot(5*x)^3*log(-5 + x) - 20*x*acot(5*x)^3*log(-5 + x))/((1 + 25*x^2)*(-5 + x))
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