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Simplificación de expresiones/ Descomposición en fracciones simples/ 1/(2*sqrt(x)*(1+x)*asin(4*x))-4*atan(sqrt(x))/(sqrt(1-16*x^2)*asin(4*x)^2)

¿Cómo vas a descomponer esta 1/(2*sqrt(x)*(1+x)*asin(4*x))-4*atan(sqrt(x))/(sqrt(1-16*x^2)*asin(4*x)^2) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                                        /  ___\      
            1                     4*atan\\/ x /      
------------------------- - -------------------------
    ___                        ___________           
2*\/ x *(1 + x)*asin(4*x)     /         2      2     
                            \/  1 - 16*x  *asin (4*x)
−4atan⁡(x)1−16x2asin⁡2(4x)+12x(x+1)asin⁡(4x)- \frac{4 \operatorname{atan}{\left(\sqrt{x} \right)}}{\sqrt{1 - 16 x^{2}} \operatorname{asin}^{2}{\left(4 x \right)}} + \frac{1}{2 \sqrt{x} \left(x + 1\right) \operatorname{asin}{\left(4 x \right)}} 
1/(((2*sqrt(x))*(1 + x))*asin(4*x)) - 4*atan(sqrt(x))/(sqrt(1 - 16*x^2)*asin(4*x)^2)
Simplificación general [src] 
   ___________                                        
  /         2                  ___             /  ___\
\/  1 - 16*x  *asin(4*x) - 8*\/ x *(1 + x)*atan\\/ x /
------------------------------------------------------
                         ___________                  
          ___           /         2      2            
      2*\/ x *(1 + x)*\/  1 - 16*x  *asin (4*x)
−8x(x+1)atan⁡(x)+1−16x2asin⁡(4x)2x1−16x2(x+1)asin⁡2(4x)\frac{- 8 \sqrt{x} \left(x + 1\right) \operatorname{atan}{\left(\sqrt{x} \right)} + \sqrt{1 - 16 x^{2}} \operatorname{asin}{\left(4 x \right)}}{2 \sqrt{x} \sqrt{1 - 16 x^{2}} \left(x + 1\right) \operatorname{asin}^{2}{\left(4 x \right)}} 
(sqrt(1 - 16*x^2)*asin(4*x) - 8*sqrt(x)*(1 + x)*atan(sqrt(x)))/(2*sqrt(x)*(1 + x)*sqrt(1 - 16*x^2)*asin(4*x)^2)
Respuesta numérica [src] 
0.5*x^(-0.5)/((1.0 + x)*asin(4*x)) - 1.0*(0.0625 - x^2)^(-0.5)*atan(sqrt(x))/asin(4*x)^2
0.5*x^(-0.5)/((1.0 + x)*asin(4*x)) - 1.0*(0.0625 - x^2)^(-0.5)*atan(sqrt(x))/asin(4*x)^2
Denominador común [src] 
   ___________                                                      
  /         2                  ___     /  ___\      3/2     /  ___\ 
\/  1 - 16*x  *asin(4*x) - 8*\/ x *atan\\/ x / - 8*x   *atan\\/ x / 
--------------------------------------------------------------------
           ___________                        ___________           
    ___   /         2      2           3/2   /         2      2     
2*\/ x *\/  1 - 16*x  *asin (4*x) + 2*x   *\/  1 - 16*x  *asin (4*x)
−8x32atan⁡(x)−8xatan⁡(x)+1−16x2asin⁡(4x)2x321−16x2asin⁡2(4x)+2x1−16x2asin⁡2(4x)\frac{- 8 x^{\frac{3}{2}} \operatorname{atan}{\left(\sqrt{x} \right)} - 8 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + \sqrt{1 - 16 x^{2}} \operatorname{asin}{\left(4 x \right)}}{2 x^{\frac{3}{2}} \sqrt{1 - 16 x^{2}} \operatorname{asin}^{2}{\left(4 x \right)} + 2 \sqrt{x} \sqrt{1 - 16 x^{2}} \operatorname{asin}^{2}{\left(4 x \right)}} 
(sqrt(1 - 16*x^2)*asin(4*x) - 8*sqrt(x)*atan(sqrt(x)) - 8*x^(3/2)*atan(sqrt(x)))/(2*sqrt(x)*sqrt(1 - 16*x^2)*asin(4*x)^2 + 2*x^(3/2)*sqrt(1 - 16*x^2)*asin(4*x)^2)
Combinatoria [src] 
   ___________                                                     
  /         2                  ___     /  ___\      3/2     /  ___\
\/  1 - 16*x  *asin(4*x) - 8*\/ x *atan\\/ x / - 8*x   *atan\\/ x /
-------------------------------------------------------------------
            ___   _______________________             2            
        2*\/ x *\/ -(1 + 4*x)*(-1 + 4*x) *(1 + x)*asin (4*x)
−8x32atan⁡(x)−8xatan⁡(x)+1−16x2asin⁡(4x)2x−(4x−1)(4x+1)(x+1)asin⁡2(4x)\frac{- 8 x^{\frac{3}{2}} \operatorname{atan}{\left(\sqrt{x} \right)} - 8 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + \sqrt{1 - 16 x^{2}} \operatorname{asin}{\left(4 x \right)}}{2 \sqrt{x} \sqrt{- \left(4 x - 1\right) \left(4 x + 1\right)} \left(x + 1\right) \operatorname{asin}^{2}{\left(4 x \right)}} 
(sqrt(1 - 16*x^2)*asin(4*x) - 8*sqrt(x)*atan(sqrt(x)) - 8*x^(3/2)*atan(sqrt(x)))/(2*sqrt(x)*sqrt(-(1 + 4*x)*(-1 + 4*x))*(1 + x)*asin(4*x)^2)
Potencias [src] 
                                        /  ___\      
            1                     4*atan\\/ x /      
------------------------- - -------------------------
  ___                          ___________           
\/ x *(2 + 2*x)*asin(4*x)     /         2      2     
                            \/  1 - 16*x  *asin (4*x)
−4atan⁡(x)1−16x2asin⁡2(4x)+1x(2x+2)asin⁡(4x)- \frac{4 \operatorname{atan}{\left(\sqrt{x} \right)}}{\sqrt{1 - 16 x^{2}} \operatorname{asin}^{2}{\left(4 x \right)}} + \frac{1}{\sqrt{x} \left(2 x + 2\right) \operatorname{asin}{\left(4 x \right)}} 
1/(sqrt(x)*(2 + 2*x)*asin(4*x)) - 4*atan(sqrt(x))/(sqrt(1 - 16*x^2)*asin(4*x)^2)
Abrimos la expresión [src] 
/       1       \                            
|---------------|                            
|    ___        |               /  ___\      
\2*\/ x *(1 + x)/         4*atan\\/ x /      
----------------- - -------------------------
    asin(4*x)          ___________           
                      /         2      2     
                    \/  1 - 16*x  *asin (4*x)
121x1x+1asin⁡(4x)−4atan⁡(x)1−16x2asin⁡2(4x)\frac{\frac{1}{2} \frac{1}{\sqrt{x}} \frac{1}{x + 1}}{\operatorname{asin}{\left(4 x \right)}} - \frac{4 \operatorname{atan}{\left(\sqrt{x} \right)}}{\sqrt{1 - 16 x^{2}} \operatorname{asin}^{2}{\left(4 x \right)}} 
(1/(2*sqrt(x)*(1 + x)))/asin(4*x) - 4*atan(sqrt(x))/(sqrt(1 - 16*x^2)*asin(4*x)^2)
Unión de expresiones racionales [src] 
   ___________                                        
  /         2                  ___             /  ___\
\/  1 - 16*x  *asin(4*x) - 8*\/ x *(1 + x)*atan\\/ x /
------------------------------------------------------
                         ___________                  
          ___           /         2      2            
      2*\/ x *(1 + x)*\/  1 - 16*x  *asin (4*x)
−8x(x+1)atan⁡(x)+1−16x2asin⁡(4x)2x1−16x2(x+1)asin⁡2(4x)\frac{- 8 \sqrt{x} \left(x + 1\right) \operatorname{atan}{\left(\sqrt{x} \right)} + \sqrt{1 - 16 x^{2}} \operatorname{asin}{\left(4 x \right)}}{2 \sqrt{x} \sqrt{1 - 16 x^{2}} \left(x + 1\right) \operatorname{asin}^{2}{\left(4 x \right)}} 
(sqrt(1 - 16*x^2)*asin(4*x) - 8*sqrt(x)*(1 + x)*atan(sqrt(x)))/(2*sqrt(x)*(1 + x)*sqrt(1 - 16*x^2)*asin(4*x)^2)
Denominador racional [src] 
                                             ___________                                   ___________                      
      2            2     2            ___   /         2                /  ___\      3/2   /         2                /  ___\
- asin (4*x) + 16*x *asin (4*x) + 8*\/ x *\/  1 - 16*x  *asin(4*x)*atan\\/ x / + 8*x   *\/  1 - 16*x  *asin(4*x)*atan\\/ x /
----------------------------------------------------------------------------------------------------------------------------
                                              ___         /         2\     3                                                
                                          2*\/ x *(1 + x)*\-1 + 16*x /*asin (4*x)
8x321−16x2asin⁡(4x)atan⁡(x)+8x1−16x2asin⁡(4x)atan⁡(x)+16x2asin⁡2(4x)−asin⁡2(4x)2x(x+1)(16x2−1)asin⁡3(4x)\frac{8 x^{\frac{3}{2}} \sqrt{1 - 16 x^{2}} \operatorname{asin}{\left(4 x \right)} \operatorname{atan}{\left(\sqrt{x} \right)} + 8 \sqrt{x} \sqrt{1 - 16 x^{2}} \operatorname{asin}{\left(4 x \right)} \operatorname{atan}{\left(\sqrt{x} \right)} + 16 x^{2} \operatorname{asin}^{2}{\left(4 x \right)} - \operatorname{asin}^{2}{\left(4 x \right)}}{2 \sqrt{x} \left(x + 1\right) \left(16 x^{2} - 1\right) \operatorname{asin}^{3}{\left(4 x \right)}} 
(-asin(4*x)^2 + 16*x^2*asin(4*x)^2 + 8*sqrt(x)*sqrt(1 - 16*x^2)*asin(4*x)*atan(sqrt(x)) + 8*x^(3/2)*sqrt(1 - 16*x^2)*asin(4*x)*atan(sqrt(x)))/(2*sqrt(x)*(1 + x)*(-1 + 16*x^2)*asin(4*x)^3)
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