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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+1+52x−1+5)10+log⁡(x2−x−1)2- \frac{\sqrt{5} \log{\left(- \frac{- 2 x + 1 + \sqrt{5}}{2 x - 1 + \sqrt{5}} \right)}}{10} + \frac{\log{\left(x^{2} - x - 1 \right)}}{2} 
log(-1 + x^2 - x)/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))
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^2 - x)/2 - sqrt(5)*log((-1 - sqrt(5) + 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
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^2 - 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
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^2 - x)/2 - sqrt(5)*log((-1 - sqrt(5) + 2*x)/(-1 + sqrt(5) + 2*x))/10
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 - 1 + \sqrt{5}} - \frac{\sqrt{5}}{2 x - 1 + \sqrt{5}} - \frac{1}{2 x - 1 + \sqrt{5}} \right)}}{10} 
(5*log(-1 + x^2 - x) - 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
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^2 - 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
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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