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Simplificación de expresiones/ Descomposición en fracciones simples/ log(x^2+x-1)/2-log((2*x-sqrt(5)+1)/(1+sqrt(5)+2*x))/(2*sqrt(5))

¿Cómo vas a descomponer esta log(x^2+x-1)/2-log((2*x-sqrt(5)+1)/(1+sqrt(5)+2*x))/(2*sqrt(5)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                     /        ___    \
                     |2*x - \/ 5  + 1|
                  log|---------------|
   / 2        \      |      ___      |
log\x  + x - 1/      \1 + \/ 5  + 2*x/
--------------- - --------------------
       2                    ___       
                        2*\/ 5
−log⁡((2x−5)+12x+(1+5))25+log⁡((x2+x)−1)2- \frac{\log{\left(\frac{\left(2 x - \sqrt{5}\right) + 1}{2 x + \left(1 + \sqrt{5}\right)} \right)}}{2 \sqrt{5}} + \frac{\log{\left(\left(x^{2} + x\right) - 1 \right)}}{2} 
log(x^2 + x - 1)/2 - log((2*x - sqrt(5) + 1)/(1 + sqrt(5) + 2*x))/(2*sqrt(5))
Simplificación general [src] 
                            /      ___      \
                     ___    |1 - \/ 5  + 2*x|
                   \/ 5 *log|---------------|
   /          2\            |      ___      |
log\-1 + x + x /            \1 + \/ 5  + 2*x/
---------------- - --------------------------
       2                       10
−5log⁡(2x−5+12x+1+5)10+log⁡(x2+x−1)2- \frac{\sqrt{5} \log{\left(\frac{2 x - \sqrt{5} + 1}{2 x + 1 + \sqrt{5}} \right)}}{10} + \frac{\log{\left(x^{2} + x - 1 \right)}}{2} 
log(-1 + x + x^2)/2 - sqrt(5)*log((1 - sqrt(5) + 2*x)/(1 + sqrt(5) + 2*x))/10
Respuesta numérica [src] 
0.5*log(x^2 + x - 1) - 0.223606797749979*log((2*x - sqrt(5) + 1)/(1 + sqrt(5) + 2*x))
0.5*log(x^2 + x - 1) - 0.223606797749979*log((2*x - sqrt(5) + 1)/(1 + sqrt(5) + 2*x))
Denominador racional [src] 
                              /                         ___                       \
     /          2\     ___    |       1               \/ 5               2*x      |
5*log\-1 + x + x / - \/ 5 *log|--------------- - --------------- + ---------------|
                              |      ___               ___               ___      |
                              \1 + \/ 5  + 2*x   1 + \/ 5  + 2*x   1 + \/ 5  + 2*x/
-----------------------------------------------------------------------------------
                                         10
5log⁡(x2+x−1)−5log⁡(2x2x+(1+5)−52x+(1+5)+12x+(1+5))10\frac{5 \log{\left(x^{2} + x - 1 \right)} - \sqrt{5} \log{\left(\frac{2 x}{2 x + \left(1 + \sqrt{5}\right)} - \frac{\sqrt{5}}{2 x + \left(1 + \sqrt{5}\right)} + \frac{1}{2 x + \left(1 + \sqrt{5}\right)} \right)}}{10} 
(5*log(-1 + x + x^2) - sqrt(5)*log(1/(1 + sqrt(5) + 2*x) - sqrt(5)/(1 + sqrt(5) + 2*x) + 2*x/(1 + sqrt(5) + 2*x)))/10
Potencias [src] 
                            /      ___      \
                     ___    |1 - \/ 5  + 2*x|
                   \/ 5 *log|---------------|
   /          2\            |      ___      |
log\-1 + x + x /            \1 + \/ 5  + 2*x/
---------------- - --------------------------
       2                       10
−5log⁡(2x−5+12x+1+5)10+log⁡(x2+x−1)2- \frac{\sqrt{5} \log{\left(\frac{2 x - \sqrt{5} + 1}{2 x + 1 + \sqrt{5}} \right)}}{10} + \frac{\log{\left(x^{2} + x - 1 \right)}}{2} 
log(-1 + x + x^2)/2 - sqrt(5)*log((1 - sqrt(5) + 2*x)/(1 + sqrt(5) + 2*x))/10
Denominador común [src] 
                            /                         ___                       \
                     ___    |       1               \/ 5               2*x      |
                   \/ 5 *log|--------------- - --------------- + ---------------|
   /          2\            |      ___               ___               ___      |
log\-1 + x + x /            \1 + \/ 5  + 2*x   1 + \/ 5  + 2*x   1 + \/ 5  + 2*x/
---------------- - --------------------------------------------------------------
       2                                         10
log⁡(x2+x−1)2−5log⁡(2x2x+1+5−52x+1+5+12x+1+5)10\frac{\log{\left(x^{2} + x - 1 \right)}}{2} - \frac{\sqrt{5} \log{\left(\frac{2 x}{2 x + 1 + \sqrt{5}} - \frac{\sqrt{5}}{2 x + 1 + \sqrt{5}} + \frac{1}{2 x + 1 + \sqrt{5}} \right)}}{10} 
log(-1 + x + x^2)/2 - sqrt(5)*log(1/(1 + sqrt(5) + 2*x) - sqrt(5)/(1 + sqrt(5) + 2*x) + 2*x/(1 + sqrt(5) + 2*x))/10
Unión de expresiones racionales [src] 
                                 /      ___      \
                          ___    |1 - \/ 5  + 2*x|
5*log(-1 + x*(1 + x)) - \/ 5 *log|---------------|
                                 |      ___      |
                                 \1 + \/ 5  + 2*x/
--------------------------------------------------
                        10
−5log⁡(2x−5+12x+1+5)+5log⁡(x(x+1)−1)10\frac{- \sqrt{5} \log{\left(\frac{2 x - \sqrt{5} + 1}{2 x + 1 + \sqrt{5}} \right)} + 5 \log{\left(x \left(x + 1\right) - 1 \right)}}{10} 
(5*log(-1 + x*(1 + x)) - sqrt(5)*log((1 - sqrt(5) + 2*x)/(1 + sqrt(5) + 2*x)))/10
Parte trigonométrica [src] 
                            /      ___      \
                     ___    |1 - \/ 5  + 2*x|
                   \/ 5 *log|---------------|
   /          2\            |      ___      |
log\-1 + x + x /            \1 + \/ 5  + 2*x/
---------------- - --------------------------
       2                       10
−5log⁡(2x−5+12x+1+5)10+log⁡(x2+x−1)2- \frac{\sqrt{5} \log{\left(\frac{2 x - \sqrt{5} + 1}{2 x + 1 + \sqrt{5}} \right)}}{10} + \frac{\log{\left(x^{2} + x - 1 \right)}}{2} 
log(-1 + x + x^2)/2 - sqrt(5)*log((1 - sqrt(5) + 2*x)/(1 + sqrt(5) + 2*x))/10
Combinatoria [src] 
                            /                         ___                       \
                     ___    |       1               \/ 5               2*x      |
                   \/ 5 *log|--------------- - --------------- + ---------------|
   /          2\            |      ___               ___               ___      |
log\-1 + x + x /            \1 + \/ 5  + 2*x   1 + \/ 5  + 2*x   1 + \/ 5  + 2*x/
---------------- - --------------------------------------------------------------
       2                                         10
log⁡(x2+x−1)2−5log⁡(2x2x+1+5−52x+1+5+12x+1+5)10\frac{\log{\left(x^{2} + x - 1 \right)}}{2} - \frac{\sqrt{5} \log{\left(\frac{2 x}{2 x + 1 + \sqrt{5}} - \frac{\sqrt{5}}{2 x + 1 + \sqrt{5}} + \frac{1}{2 x + 1 + \sqrt{5}} \right)}}{10} 
log(-1 + x + x^2)/2 - sqrt(5)*log(1/(1 + sqrt(5) + 2*x) - sqrt(5)/(1 + sqrt(5) + 2*x) + 2*x/(1 + sqrt(5) + 2*x))/10
Compilar la expresión [src] 
                           /        ___    \
                    ___    |2*x - \/ 5  + 1|
                  \/ 5 *log|---------------|
   / 2        \            |      ___      |
log\x  + x - 1/            \1 + \/ 5  + 2*x/
--------------- - --------------------------
       2                      10
−5log⁡((2x−5)+12x+(1+5))10+log⁡((x2+x)−1)2- \frac{\sqrt{5} \log{\left(\frac{\left(2 x - \sqrt{5}\right) + 1}{2 x + \left(1 + \sqrt{5}\right)} \right)}}{10} + \frac{\log{\left(\left(x^{2} + x\right) - 1 \right)}}{2} 
log(x^2 + x - 1)/2 - sqrt(5)*log((2*x - sqrt(5) + 1)/(1 + sqrt(5) + 2*x))/10
Abrimos la expresión [src] 
                           /        ___    \
                    ___    |2*x - \/ 5  + 1|
                  \/ 5 *log|---------------|
   / 2        \            |      ___      |
log\x  + x - 1/            \1 + \/ 5  + 2*x/
--------------- - --------------------------
       2                      10
−5log⁡((2x−5)+12x+(1+5))10+log⁡((x2+x)−1)2- \frac{\sqrt{5} \log{\left(\frac{\left(2 x - \sqrt{5}\right) + 1}{2 x + \left(1 + \sqrt{5}\right)} \right)}}{10} + \frac{\log{\left(\left(x^{2} + x\right) - 1 \right)}}{2} 
log(x^2 + x - 1)/2 - sqrt(5)*log((2*x - sqrt(5) + 1)/(1 + sqrt(5) + 2*x))/10
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