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Simplificación de expresiones/ Descomposición en fracciones simples/ acot(3*x)^4/(x-4)-12*acot(3*x)^3*log(x-4)/(1+9*x^2)

¿Cómo vas a descomponer esta acot(3*x)^4/(x-4)-12*acot(3*x)^3*log(x-4)/(1+9*x^2) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    4               3                
acot (3*x)   12*acot (3*x)*log(x - 4)
---------- - ------------------------
  x - 4                     2        
                     1 + 9*x
$$- \frac{\log{\left(x - 4 \right)} 12 \operatorname{acot}^{3}{\left(3 x \right)}}{9 x^{2} + 1} + \frac{\operatorname{acot}^{4}{\left(3 x \right)}}{x - 4}$$ 
acot(3*x)^4/(x - 4) - (12*acot(3*x)^3)*log(x - 4)/(1 + 9*x^2)
Simplificación general [src] 
    3      //       2\                                    \
acot (3*x)*\\1 + 9*x /*acot(3*x) - 12*(-4 + x)*log(-4 + x)/
-----------------------------------------------------------
                    /       2\                             
                    \1 + 9*x /*(-4 + x)
$$\frac{\left(- 12 \left(x - 4\right) \log{\left(x - 4 \right)} + \left(9 x^{2} + 1\right) \operatorname{acot}{\left(3 x \right)}\right) \operatorname{acot}^{3}{\left(3 x \right)}}{\left(x - 4\right) \left(9 x^{2} + 1\right)}$$ 
acot(3*x)^3*((1 + 9*x^2)*acot(3*x) - 12*(-4 + x)*log(-4 + x))/((1 + 9*x^2)*(-4 + x))
Respuesta numérica [src] 
acot(3*x)^4/(-4.0 + x) - 12.0*acot(3*x)^3*log(x - 4)/(1.0 + 9.0*x^2)
acot(3*x)^4/(-4.0 + x) - 12.0*acot(3*x)^3*log(x - 4)/(1.0 + 9.0*x^2)
Combinatoria [src] 
    3      /                                       2                      \
acot (3*x)*\48*log(-4 + x) - 12*x*log(-4 + x) + 9*x *acot(3*x) + acot(3*x)/
---------------------------------------------------------------------------
                            /       2\                                     
                            \1 + 9*x /*(-4 + x)
$$\frac{\left(9 x^{2} \operatorname{acot}{\left(3 x \right)} - 12 x \log{\left(x - 4 \right)} + 48 \log{\left(x - 4 \right)} + \operatorname{acot}{\left(3 x \right)}\right) \operatorname{acot}^{3}{\left(3 x \right)}}{\left(x - 4\right) \left(9 x^{2} + 1\right)}$$ 
acot(3*x)^3*(48*log(-4 + x) - 12*x*log(-4 + x) + 9*x^2*acot(3*x) + acot(3*x))/((1 + 9*x^2)*(-4 + x))
Parte trigonométrica [src] 
    4               3                 
acot (3*x)   12*acot (3*x)*log(-4 + x)
---------- - -------------------------
  -4 + x                     2        
                      1 + 9*x
$$- \frac{12 \log{\left(x - 4 \right)} \operatorname{acot}^{3}{\left(3 x \right)}}{9 x^{2} + 1} + \frac{\operatorname{acot}^{4}{\left(3 x \right)}}{x - 4}$$ 
acot(3*x)^4/(-4 + x) - 12*acot(3*x)^3*log(-4 + x)/(1 + 9*x^2)
Denominador común [src] 
    4           2     4               3                             3                 
acot (3*x) + 9*x *acot (3*x) + 48*acot (3*x)*log(-4 + x) - 12*x*acot (3*x)*log(-4 + x)
--------------------------------------------------------------------------------------
                                             2      3                                 
                                -4 + x - 36*x  + 9*x
$$\frac{9 x^{2} \operatorname{acot}^{4}{\left(3 x \right)} - 12 x \log{\left(x - 4 \right)} \operatorname{acot}^{3}{\left(3 x \right)} + 48 \log{\left(x - 4 \right)} \operatorname{acot}^{3}{\left(3 x \right)} + \operatorname{acot}^{4}{\left(3 x \right)}}{9 x^{3} - 36 x^{2} + x - 4}$$ 
(acot(3*x)^4 + 9*x^2*acot(3*x)^4 + 48*acot(3*x)^3*log(-4 + x) - 12*x*acot(3*x)^3*log(-4 + x))/(-4 + x - 36*x^2 + 9*x^3)
Unión de expresiones racionales [src] 
    3      //       2\                                    \
acot (3*x)*\\1 + 9*x /*acot(3*x) - 12*(-4 + x)*log(-4 + x)/
-----------------------------------------------------------
                    /       2\                             
                    \1 + 9*x /*(-4 + x)
$$\frac{\left(- 12 \left(x - 4\right) \log{\left(x - 4 \right)} + \left(9 x^{2} + 1\right) \operatorname{acot}{\left(3 x \right)}\right) \operatorname{acot}^{3}{\left(3 x \right)}}{\left(x - 4\right) \left(9 x^{2} + 1\right)}$$ 
acot(3*x)^3*((1 + 9*x^2)*acot(3*x) - 12*(-4 + x)*log(-4 + x))/((1 + 9*x^2)*(-4 + x))
Denominador racional [src] 
    4           2     4               3                             3                 
acot (3*x) + 9*x *acot (3*x) + 48*acot (3*x)*log(-4 + x) - 12*x*acot (3*x)*log(-4 + x)
--------------------------------------------------------------------------------------
                                 /       2\                                           
                                 \1 + 9*x /*(-4 + x)
$$\frac{9 x^{2} \operatorname{acot}^{4}{\left(3 x \right)} - 12 x \log{\left(x - 4 \right)} \operatorname{acot}^{3}{\left(3 x \right)} + 48 \log{\left(x - 4 \right)} \operatorname{acot}^{3}{\left(3 x \right)} + \operatorname{acot}^{4}{\left(3 x \right)}}{\left(x - 4\right) \left(9 x^{2} + 1\right)}$$ 
(acot(3*x)^4 + 9*x^2*acot(3*x)^4 + 48*acot(3*x)^3*log(-4 + x) - 12*x*acot(3*x)^3*log(-4 + x))/((1 + 9*x^2)*(-4 + x))
Potencias [src] 
    4               3                 
acot (3*x)   12*acot (3*x)*log(-4 + x)
---------- - -------------------------
  -4 + x                     2        
                      1 + 9*x
$$- \frac{12 \log{\left(x - 4 \right)} \operatorname{acot}^{3}{\left(3 x \right)}}{9 x^{2} + 1} + \frac{\operatorname{acot}^{4}{\left(3 x \right)}}{x - 4}$$ 
acot(3*x)^4/(-4 + x) - 12*acot(3*x)^3*log(-4 + x)/(1 + 9*x^2)
Compilar la expresión [src] 
    4               3                
acot (3*x)   12*acot (3*x)*log(x - 4)
---------- - ------------------------
  -4 + x                    2        
                     1 + 9*x
$$- \frac{12 \log{\left(x - 4 \right)} \operatorname{acot}^{3}{\left(3 x \right)}}{9 x^{2} + 1} + \frac{\operatorname{acot}^{4}{\left(3 x \right)}}{x - 4}$$ 
acot(3*x)^4/(-4 + x) - 12*acot(3*x)^3*log(x - 4)/(1 + 9*x^2)
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