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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    3/ -x\                 -x     2/ -x\       
atan \2  /*(-1 - 2*x)   3*2  *atan \2  /*log(2)
--------------------- - -----------------------
              2           /     -2*x\ / 2    \ 
      / 2    \            \1 + 2    /*\x  + x/ 
      \x  + x/
(−2x−1)atan⁡3(2−x)(x2+x)2−3⋅2−xatan⁡2(2−x)log⁡(2)(1+2−2x)(x2+x)\frac{\left(- 2 x - 1\right) \operatorname{atan}^{3}{\left(2^{- x} \right)}}{\left(x^{2} + x\right)^{2}} - \frac{3 \cdot 2^{- x} \operatorname{atan}^{2}{\left(2^{- x} \right)} \log{\left(2 \right)}}{\left(1 + 2^{- 2 x}\right) \left(x^{2} + x\right)} 
(atan(2^(-x))^3*(-1 - 2*x))/(x^2 + x)^2 - ((3*2^(-x))*atan(2^(-x))^2)*log(2)/((1 + 2^(-2*x))*(x^2 + x))
Simplificación general [src] 
     2/ -x\ //     x\               / -x\      x               \ 
-atan \2  /*\\1 + 4 /*(1 + 2*x)*atan\2  / + x*2 *(1 + x)*log(8)/ 
-----------------------------------------------------------------
                       2        2 /     2*x\                     
                      x *(1 + x) *\1 + 2   /
−(2xx(x+1)log⁡(8)+(4x+1)(2x+1)atan⁡(2−x))atan⁡2(2−x)x2(22x+1)(x+1)2- \frac{\left(2^{x} x \left(x + 1\right) \log{\left(8 \right)} + \left(4^{x} + 1\right) \left(2 x + 1\right) \operatorname{atan}{\left(2^{- x} \right)}\right) \operatorname{atan}^{2}{\left(2^{- x} \right)}}{x^{2} \left(2^{2 x} + 1\right) \left(x + 1\right)^{2}} 
-atan(2^(-x))^2*((1 + 4^x)*(1 + 2*x)*atan(2^(-x)) + x*2^x*(1 + x)*log(8))/(x^2*(1 + x)^2*(1 + 2^(2*x)))
Respuesta numérica [src] 
atan(2^(-x))^3*(-1.0 - 2.0*x)/(x + x^2)^2 - 2.07944154167984*2.0^(-x)*atan(2^(-x))^2/((1.0 + 2.0^(-2.0*x))*(x + x^2))
atan(2^(-x))^3*(-1.0 - 2.0*x)/(x + x^2)^2 - 2.07944154167984*2.0^(-x)*atan(2^(-x))^2/((1.0 + 2.0^(-2.0*x))*(x + x^2))
Potencias [src] 
    3/ -x\                 -x     2/ -x\       
atan \2  /*(-1 - 2*x)   3*2  *atan \2  /*log(2)
--------------------- - -----------------------
              2           /     -2*x\ /     2\ 
      /     2\            \1 + 2    /*\x + x / 
      \x + x /
(−2x−1)atan⁡3(2−x)(x2+x)2−3⋅2−xlog⁡(2)atan⁡2(2−x)(1+2−2x)(x2+x)\frac{\left(- 2 x - 1\right) \operatorname{atan}^{3}{\left(2^{- x} \right)}}{\left(x^{2} + x\right)^{2}} - \frac{3 \cdot 2^{- x} \log{\left(2 \right)} \operatorname{atan}^{2}{\left(2^{- x} \right)}}{\left(1 + 2^{- 2 x}\right) \left(x^{2} + x\right)} 
    3/ -x\                 -x     2/ -x\       
atan \2  /*(-1 - 2*x)   3*2  *atan \2  /*log(2)
--------------------- - -----------------------
              2            /     -x\ /     2\  
      /     2\             \1 + 4  /*\x + x /  
      \x + x /
(−2x−1)atan⁡3(2−x)(x2+x)2−3⋅2−xlog⁡(2)atan⁡2(2−x)(1+4−x)(x2+x)\frac{\left(- 2 x - 1\right) \operatorname{atan}^{3}{\left(2^{- x} \right)}}{\left(x^{2} + x\right)^{2}} - \frac{3 \cdot 2^{- x} \log{\left(2 \right)} \operatorname{atan}^{2}{\left(2^{- x} \right)}}{\left(1 + 4^{- x}\right) \left(x^{2} + x\right)} 
atan(2^(-x))^3*(-1 - 2*x)/(x + x^2)^2 - 3*2^(-x)*atan(2^(-x))^2*log(2)/((1 + 4^(-x))*(x + x^2))
Combinatoria [src] 
     2/ -x\ / 2*x     / -x\           / -x\        2*x     / -x\        x             x  2              / -x\\ 
-atan \2  /*\2   *atan\2  / + 2*x*atan\2  / + 2*x*2   *atan\2  / + 3*x*2 *log(2) + 3*2 *x *log(2) + atan\2  // 
---------------------------------------------------------------------------------------------------------------
                                              2        2 /     2*x\                                            
                                             x *(1 + x) *\1 + 2   /
−(2⋅22xxatan⁡(2−x)+22xatan⁡(2−x)+3⋅2xx2log⁡(2)+3⋅2xxlog⁡(2)+2xatan⁡(2−x)+atan⁡(2−x))atan⁡2(2−x)x2(22x+1)(x+1)2- \frac{\left(2 \cdot 2^{2 x} x \operatorname{atan}{\left(2^{- x} \right)} + 2^{2 x} \operatorname{atan}{\left(2^{- x} \right)} + 3 \cdot 2^{x} x^{2} \log{\left(2 \right)} + 3 \cdot 2^{x} x \log{\left(2 \right)} + 2 x \operatorname{atan}{\left(2^{- x} \right)} + \operatorname{atan}{\left(2^{- x} \right)}\right) \operatorname{atan}^{2}{\left(2^{- x} \right)}}{x^{2} \left(2^{2 x} + 1\right) \left(x + 1\right)^{2}} 
-atan(2^(-x))^2*(2^(2*x)*atan(2^(-x)) + 2*x*atan(2^(-x)) + 2*x*2^(2*x)*atan(2^(-x)) + 3*x*2^x*log(2) + 3*2^x*x^2*log(2) + atan(2^(-x)))/(x^2*(1 + x)^2*(1 + 2^(2*x)))
Denominador racional [src] 
    /                                                                                                                                               2                  \
 -x |     x     3/ -x\      3*x     3/ -x\      x  2     3/ -x\      3*x  2     3/ -x\      x  3     3/ -x\      3*x  3     3/ -x\      2*x /     2\      2/ -x\       |
2  *\- x*2 *atan \2  / - x*2   *atan \2  / - 3*2 *x *atan \2  / - 3*2   *x *atan \2  / - 2*2 *x *atan \2  / - 2*2   *x *atan \2  / - 3*2   *\x + x / *atan \2  /*log(2)/
------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                             3                                                                          
                                                                          /     2*x\ /     2\                                                                           
                                                                          \1 + 2   /*\x + x /
2−x(−2⋅23xx3atan⁡3((12)x)−3⋅23xx2atan⁡3((12)x)−23xxatan⁡3((12)x)−3⋅22x(x2+x)2log⁡(2)atan⁡2((12)x)−2⋅2xx3atan⁡3((12)x)−3⋅2xx2atan⁡3((12)x)−2xxatan⁡3((12)x))(22x+1)(x2+x)3\frac{2^{- x} \left(- 2 \cdot 2^{3 x} x^{3} \operatorname{atan}^{3}{\left(\left(\frac{1}{2}\right)^{x} \right)} - 3 \cdot 2^{3 x} x^{2} \operatorname{atan}^{3}{\left(\left(\frac{1}{2}\right)^{x} \right)} - 2^{3 x} x \operatorname{atan}^{3}{\left(\left(\frac{1}{2}\right)^{x} \right)} - 3 \cdot 2^{2 x} \left(x^{2} + x\right)^{2} \log{\left(2 \right)} \operatorname{atan}^{2}{\left(\left(\frac{1}{2}\right)^{x} \right)} - 2 \cdot 2^{x} x^{3} \operatorname{atan}^{3}{\left(\left(\frac{1}{2}\right)^{x} \right)} - 3 \cdot 2^{x} x^{2} \operatorname{atan}^{3}{\left(\left(\frac{1}{2}\right)^{x} \right)} - 2^{x} x \operatorname{atan}^{3}{\left(\left(\frac{1}{2}\right)^{x} \right)}\right)}{\left(2^{2 x} + 1\right) \left(x^{2} + x\right)^{3}} 
2^(-x)*(-x*2^x*atan((1/2)^x)^3 - x*2^(3*x)*atan((1/2)^x)^3 - 3*2^x*x^2*atan((1/2)^x)^3 - 3*2^(3*x)*x^2*atan((1/2)^x)^3 - 2*2^x*x^3*atan((1/2)^x)^3 - 2*2^(3*x)*x^3*atan((1/2)^x)^3 - 3*2^(2*x)*(x + x^2)^2*atan((1/2)^x)^2*log(2))/((1 + 2^(2*x))*(x + x^2)^3)
Denominador común [src] 
 /    3/ -x\    2*x     3/ -x\           3/ -x\        2*x     3/ -x\        x     2/ -x\             x  2     2/ -x\       \ 
-\atan \2  / + 2   *atan \2  / + 2*x*atan \2  / + 2*x*2   *atan \2  / + 3*x*2 *atan \2  /*log(2) + 3*2 *x *atan \2  /*log(2)/ 
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                                         2    4      3    2*x  2    2*x  4      2*x  3                                        
                                        x  + x  + 2*x  + 2   *x  + 2   *x  + 2*2   *x
−2⋅22xxatan⁡3(2−x)+22xatan⁡3(2−x)+3⋅2xx2log⁡(2)atan⁡2(2−x)+3⋅2xxlog⁡(2)atan⁡2(2−x)+2xatan⁡3(2−x)+atan⁡3(2−x)22xx4+2⋅22xx3+22xx2+x4+2x3+x2- \frac{2 \cdot 2^{2 x} x \operatorname{atan}^{3}{\left(2^{- x} \right)} + 2^{2 x} \operatorname{atan}^{3}{\left(2^{- x} \right)} + 3 \cdot 2^{x} x^{2} \log{\left(2 \right)} \operatorname{atan}^{2}{\left(2^{- x} \right)} + 3 \cdot 2^{x} x \log{\left(2 \right)} \operatorname{atan}^{2}{\left(2^{- x} \right)} + 2 x \operatorname{atan}^{3}{\left(2^{- x} \right)} + \operatorname{atan}^{3}{\left(2^{- x} \right)}}{2^{2 x} x^{4} + 2 \cdot 2^{2 x} x^{3} + 2^{2 x} x^{2} + x^{4} + 2 x^{3} + x^{2}} 
-(atan(2^(-x))^3 + 2^(2*x)*atan(2^(-x))^3 + 2*x*atan(2^(-x))^3 + 2*x*2^(2*x)*atan(2^(-x))^3 + 3*x*2^x*atan(2^(-x))^2*log(2) + 3*2^x*x^2*atan(2^(-x))^2*log(2))/(x^2 + x^4 + 2*x^3 + 2^(2*x)*x^2 + 2^(2*x)*x^4 + 2*2^(2*x)*x^3)
Unión de expresiones racionales [src] 
    2/ -x\ //     2*x\                / -x\        x               \
atan \2  /*\\1 + 2   /*(-1 - 2*x)*atan\2  / - 3*x*2 *(1 + x)*log(2)/
--------------------------------------------------------------------
                        2        2 /     2*x\                       
                       x *(1 + x) *\1 + 2   /
(−3⋅2xx(x+1)log⁡(2)+(22x+1)(−2x−1)atan⁡(2−x))atan⁡2(2−x)x2(22x+1)(x+1)2\frac{\left(- 3 \cdot 2^{x} x \left(x + 1\right) \log{\left(2 \right)} + \left(2^{2 x} + 1\right) \left(- 2 x - 1\right) \operatorname{atan}{\left(2^{- x} \right)}\right) \operatorname{atan}^{2}{\left(2^{- x} \right)}}{x^{2} \left(2^{2 x} + 1\right) \left(x + 1\right)^{2}} 
atan(2^(-x))^2*((1 + 2^(2*x))*(-1 - 2*x)*atan(2^(-x)) - 3*x*2^x*(1 + x)*log(2))/(x^2*(1 + x)^2*(1 + 2^(2*x)))
Parte trigonométrica [src] 
    3/ -x\                 -x     2/ -x\       
atan \2  /*(-1 - 2*x)   3*2  *atan \2  /*log(2)
--------------------- - -----------------------
              2           /     -2*x\ /     2\ 
      /     2\            \1 + 2    /*\x + x / 
      \x + x /
(−2x−1)atan⁡3(2−x)(x2+x)2−3⋅2−xlog⁡(2)atan⁡2(2−x)(1+2−2x)(x2+x)\frac{\left(- 2 x - 1\right) \operatorname{atan}^{3}{\left(2^{- x} \right)}}{\left(x^{2} + x\right)^{2}} - \frac{3 \cdot 2^{- x} \log{\left(2 \right)} \operatorname{atan}^{2}{\left(2^{- x} \right)}}{\left(1 + 2^{- 2 x}\right) \left(x^{2} + x\right)} 
atan(2^(-x))^3*(-1 - 2*x)/(x + x^2)^2 - 3*2^(-x)*atan(2^(-x))^2*log(2)/((1 + 2^(-2*x))*(x + x^2))
Abrimos la expresión [src] 
    3/ -x\                 -x     2/ -x\       
atan \2  /*(-1 - 2*x)   3*2  *atan \2  /*log(2)
--------------------- - -----------------------
              2           /     -2*x\ / 2    \ 
      / 2    \            \1 + 2    /*\x  + x/ 
      \x  + x/
(−2x−1)atan⁡3(2−x)(x2+x)2−3⋅2−xlog⁡(2)atan⁡2(2−x)(1+2−2x)(x2+x)\frac{\left(- 2 x - 1\right) \operatorname{atan}^{3}{\left(2^{- x} \right)}}{\left(x^{2} + x\right)^{2}} - \frac{3 \cdot 2^{- x} \log{\left(2 \right)} \operatorname{atan}^{2}{\left(2^{- x} \right)}}{\left(1 + 2^{- 2 x}\right) \left(x^{2} + x\right)} 
(atan(2^(-x))^3*(-1 - 2*x))/(x^2 + x)^2 - 3*2^(-x)*atan(2^(-x))^2*log(2)/((1 + 2^(-2*x))*(x^2 + x))
Compilar la expresión [src] 
    3/ -x\                 -x     2/ -x\       
atan \2  /*(-1 - 2*x)   3*2  *atan \2  /*log(2)
--------------------- - -----------------------
              2           /     -2*x\ /     2\ 
      /     2\            \1 + 2    /*\x + x / 
      \x + x /
(−2x−1)atan⁡3(2−x)(x2+x)2−3⋅2−xlog⁡(2)atan⁡2(2−x)(1+2−2x)(x2+x)\frac{\left(- 2 x - 1\right) \operatorname{atan}^{3}{\left(2^{- x} \right)}}{\left(x^{2} + x\right)^{2}} - \frac{3 \cdot 2^{- x} \log{\left(2 \right)} \operatorname{atan}^{2}{\left(2^{- x} \right)}}{\left(1 + 2^{- 2 x}\right) \left(x^{2} + x\right)} 
atan(2^(-x))^3*(-1 - 2*x)/(x + x^2)^2 - 3*2^(-x)*atan(2^(-x))^2*log(2)/((1 + 2^(-2*x))*(x + x^2))
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