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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   / 2    \                                              
log\x  + 1/   log(x + 1)   atan(x)   log(x - 1)      1   
----------- + ---------- + ------- + ---------- - -------
     4            4           2          4        2*x - 2
$$\left(\left(\left(\frac{\log{\left(x + 1 \right)}}{4} + \frac{\log{\left(x^{2} + 1 \right)}}{4}\right) + \frac{\operatorname{atan}{\left(x \right)}}{2}\right) + \frac{\log{\left(x - 1 \right)}}{4}\right) - \frac{1}{2 x - 2}$$ 
log(x^2 + 1)/4 + log(x + 1)/4 + atan(x)/2 + log(x - 1)/4 - 1/(2*x - 2)
Simplificación general [src] 
              /                            /     2\              \
-2 + (-1 + x)*\2*atan(x) + log(1 + x) + log\1 + x / + log(-1 + x)/
------------------------------------------------------------------
                            4*(-1 + x)
$$\frac{\left(x - 1\right) \left(\log{\left(x - 1 \right)} + \log{\left(x + 1 \right)} + \log{\left(x^{2} + 1 \right)} + 2 \operatorname{atan}{\left(x \right)}\right) - 2}{4 \left(x - 1\right)}$$ 
(-2 + (-1 + x)*(2*atan(x) + log(1 + x) + log(1 + x^2) + log(-1 + x)))/(4*(-1 + x))
Respuesta numérica [src] 
-1/(-2.0 + 2.0*x) + 0.25*log(x + 1) + 0.25*log(x - 1) + 0.25*log(x^2 + 1) + 0.5*atan(x)
-1/(-2.0 + 2.0*x) + 0.25*log(x + 1) + 0.25*log(x - 1) + 0.25*log(x^2 + 1) + 0.5*atan(x)
Parte trigonométrica [src] 
                                     /     2\              
atan(x)      1       log(1 + x)   log\1 + x /   log(-1 + x)
------- - -------- + ---------- + ----------- + -----------
   2      -2 + 2*x       4             4             4
$$\frac{\log{\left(x - 1 \right)}}{4} + \frac{\log{\left(x + 1 \right)}}{4} + \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2} - \frac{1}{2 x - 2}$$ 
atan(x)/2 - 1/(-2 + 2*x) + log(1 + x)/4 + log(1 + x^2)/4 + log(-1 + x)/4
Unión de expresiones racionales [src] 
              /                            /     2\              \
-2 + (-1 + x)*\2*atan(x) + log(1 + x) + log\1 + x / + log(-1 + x)/
------------------------------------------------------------------
                            4*(-1 + x)
$$\frac{\left(x - 1\right) \left(\log{\left(x - 1 \right)} + \log{\left(x + 1 \right)} + \log{\left(x^{2} + 1 \right)} + 2 \operatorname{atan}{\left(x \right)}\right) - 2}{4 \left(x - 1\right)}$$ 
(-2 + (-1 + x)*(2*atan(x) + log(1 + x) + log(1 + x^2) + log(-1 + x)))/(4*(-1 + x))
Potencias [src] 
                                     /     2\              
atan(x)      1       log(1 + x)   log\1 + x /   log(-1 + x)
------- - -------- + ---------- + ----------- + -----------
   2      -2 + 2*x       4             4             4
$$\frac{\log{\left(x - 1 \right)}}{4} + \frac{\log{\left(x + 1 \right)}}{4} + \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2} - \frac{1}{2 x - 2}$$ 
atan(x)/2 - 1/(-2 + 2*x) + log(1 + x)/4 + log(1 + x^2)/4 + log(-1 + x)/4
Combinatoria [src] 
                     /     2\                                                 /     2\                              
-2 - log(1 + x) - log\1 + x / - log(-1 + x) - 2*atan(x) + x*log(1 + x) + x*log\1 + x / + x*log(-1 + x) + 2*x*atan(x)
--------------------------------------------------------------------------------------------------------------------
                                                     4*(-1 + x)
$$\frac{x \log{\left(x - 1 \right)} + x \log{\left(x + 1 \right)} + x \log{\left(x^{2} + 1 \right)} + 2 x \operatorname{atan}{\left(x \right)} - \log{\left(x - 1 \right)} - \log{\left(x + 1 \right)} - \log{\left(x^{2} + 1 \right)} - 2 \operatorname{atan}{\left(x \right)} - 2}{4 \left(x - 1\right)}$$ 
(-2 - log(1 + x) - log(1 + x^2) - log(-1 + x) - 2*atan(x) + x*log(1 + x) + x*log(1 + x^2) + x*log(-1 + x) + 2*x*atan(x))/(4*(-1 + x))
Denominador racional [src] 
                     /     2\                                                 /     2\                              
-2 - log(1 + x) - log\1 + x / - log(-1 + x) - 2*atan(x) + x*log(1 + x) + x*log\1 + x / + x*log(-1 + x) + 2*x*atan(x)
--------------------------------------------------------------------------------------------------------------------
                                                      -4 + 4*x
$$\frac{x \log{\left(x - 1 \right)} + x \log{\left(x + 1 \right)} + x \log{\left(x^{2} + 1 \right)} + 2 x \operatorname{atan}{\left(x \right)} - \log{\left(x - 1 \right)} - \log{\left(x + 1 \right)} - \log{\left(x^{2} + 1 \right)} - 2 \operatorname{atan}{\left(x \right)} - 2}{4 x - 4}$$ 
(-2 - log(1 + x) - log(1 + x^2) - log(-1 + x) - 2*atan(x) + x*log(1 + x) + x*log(1 + x^2) + x*log(-1 + x) + 2*x*atan(x))/(-4 + 4*x)
Compilar la expresión [src] 
                                                  / 2    \
atan(x)      1       log(x + 1)   log(x - 1)   log\x  + 1/
------- - -------- + ---------- + ---------- + -----------
   2      -2 + 2*x       4            4             4
$$\frac{\log{\left(x - 1 \right)}}{4} + \frac{\log{\left(x + 1 \right)}}{4} + \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2} - \frac{1}{2 x - 2}$$ 
atan(x)/2 - 1/(-2 + 2*x) + log(x + 1)/4 + log(x - 1)/4 + log(x^2 + 1)/4
Denominador común [src] 
                                     /     2\              
atan(x)      1       log(1 + x)   log\1 + x /   log(-1 + x)
------- - -------- + ---------- + ----------- + -----------
   2      -2 + 2*x       4             4             4
$$\frac{\log{\left(x - 1 \right)}}{4} + \frac{\log{\left(x + 1 \right)}}{4} + \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2} - \frac{1}{2 x - 2}$$ 
atan(x)/2 - 1/(-2 + 2*x) + log(1 + x)/4 + log(1 + x^2)/4 + log(-1 + x)/4
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