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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   /    1\              
log|E + -|              
   \    x/        1     
---------- - -----------
    2            /    1\
             2*x*|E + -|
                 \    x/
$$\frac{\log{\left(e + \frac{1}{x} \right)}}{2} - \frac{1}{2 x \left(e + \frac{1}{x}\right)}$$ 
log(E + 1/x)/2 - 1/((2*x)*(E + 1/x))
Simplificación general [src] 
                  /1 + E*x\
-1 + (1 + E*x)*log|-------|
                  \   x   /
---------------------------
        2*(1 + E*x)
$$\frac{\left(e x + 1\right) \log{\left(\frac{e x + 1}{x} \right)} - 1}{2 \left(e x + 1\right)}$$ 
(-1 + (1 + E*x)*log((1 + E*x)/x))/(2*(1 + E*x))
Respuesta numérica [src] 
0.5*log(E + 1/x) - 0.5/(x*(2.71828182845905 + 1/x))
0.5*log(E + 1/x) - 0.5/(x*(2.71828182845905 + 1/x))
Denominador común [src] 
   /    1\            
log|E + -|            
   \    x/       1    
---------- - ---------
    2        2 + 2*E*x
$$\frac{\log{\left(e + \frac{1}{x} \right)}}{2} - \frac{1}{2 e x + 2}$$ 
log(E + 1/x)/2 - 1/(2 + 2*E*x)
Denominador racional [src] 
                        /    1\
-2*x + 2*x*(1 + E*x)*log|E + -|
                        \    x/
-------------------------------
         4*x*(1 + E*x)
$$\frac{2 x \left(e x + 1\right) \log{\left(e + \frac{1}{x} \right)} - 2 x}{4 x \left(e x + 1\right)}$$ 
(-2*x + 2*x*(1 + E*x)*log(E + 1/x))/(4*x*(1 + E*x))
Unión de expresiones racionales [src] 
                  /1 + E*x\
-1 + (1 + E*x)*log|-------|
                  \   x   /
---------------------------
        2*(1 + E*x)
$$\frac{\left(e x + 1\right) \log{\left(\frac{e x + 1}{x} \right)} - 1}{2 \left(e x + 1\right)}$$ 
(-1 + (1 + E*x)*log((1 + E*x)/x))/(2*(1 + E*x))
Combinatoria [src] 
            /    1\      /    1\
-1 + E*x*log|E + -| + log|E + -|
            \    x/      \    x/
--------------------------------
          2*(1 + E*x)
$$\frac{e x \log{\left(e + \frac{1}{x} \right)} + \log{\left(e + \frac{1}{x} \right)} - 1}{2 \left(e x + 1\right)}$$ 
(-1 + E*x*log(E + 1/x) + log(E + 1/x))/(2*(1 + E*x))
Parte trigonométrica [src] 
   /1                    \                              
log|- + cosh(1) + sinh(1)|                              
   \x                    /                1             
-------------------------- - ---------------------------
            2                    /1                    \
                             2*x*|- + cosh(1) + sinh(1)|
                                 \x                    /
$$\frac{\log{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)} + \frac{1}{x} \right)}}{2} - \frac{1}{2 x \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)} + \frac{1}{x}\right)}$$ 
log(1/x + cosh(1) + sinh(1))/2 - 1/(2*x*(1/x + cosh(1) + sinh(1)))
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