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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   /     __________    \      /     __________    \
   |    /      1       |      |    /      1       |
log|   /  1 - ----  + 1|   log|   /  1 - ----  - 1|
   |  /        2*x     |      |  /        2*x     |
   \\/        E        /      \\/        E        /
------------------------ - ------------------------
        2*log(E)                   2*log(E)
−log⁡(1−1e2x−1)2log⁡(e)+log⁡(1−1e2x+1)2log⁡(e)- \frac{\log{\left(\sqrt{1 - \frac{1}{e^{2 x}}} - 1 \right)}}{2 \log{\left(e \right)}} + \frac{\log{\left(\sqrt{1 - \frac{1}{e^{2 x}}} + 1 \right)}}{2 \log{\left(e \right)}} 
log(sqrt(1 - 1/E^(2*x)) + 1)/((2*log(E))) - log(sqrt(1 - 1/E^(2*x)) - 1)/(2*log(E))
Simplificación general [src] 
   /       ___________\      /        ___________\
   |      /      -2*x |      |       /      -2*x |
log\1 + \/  1 - e     /   log\-1 + \/  1 - e     /
----------------------- - ------------------------
           2                         2
−log⁡(1−e−2x−1)2+log⁡(1−e−2x+1)2- \frac{\log{\left(\sqrt{1 - e^{- 2 x}} - 1 \right)}}{2} + \frac{\log{\left(\sqrt{1 - e^{- 2 x}} + 1 \right)}}{2} 
log(1 + sqrt(1 - exp(-2*x)))/2 - log(-1 + sqrt(1 - exp(-2*x)))/2
Respuesta numérica [src] 
0.5*log(sqrt(1 - 1/E^(2*x)) + 1) - 0.5*log(sqrt(1 - 1/E^(2*x)) - 1)
0.5*log(sqrt(1 - 1/E^(2*x)) + 1) - 0.5*log(sqrt(1 - 1/E^(2*x)) - 1)
Combinatoria [src] 
   /       ___________\      /        ___________\
   |      /      -2*x |      |       /      -2*x |
log\1 + \/  1 - e     /   log\-1 + \/  1 - e     /
----------------------- - ------------------------
           2                         2
−log⁡(1−e−2x−1)2+log⁡(1−e−2x+1)2- \frac{\log{\left(\sqrt{1 - e^{- 2 x}} - 1 \right)}}{2} + \frac{\log{\left(\sqrt{1 - e^{- 2 x}} + 1 \right)}}{2} 
log(1 + sqrt(1 - exp(-2*x)))/2 - log(-1 + sqrt(1 - exp(-2*x)))/2
Denominador común [src] 
   /       ___________\      /        ___________\
   |      /      -2*x |      |       /      -2*x |
log\1 + \/  1 - e     /   log\-1 + \/  1 - e     /
----------------------- - ------------------------
           2                         2
−log⁡(1−e−2x−1)2+log⁡(1−e−2x+1)2- \frac{\log{\left(\sqrt{1 - e^{- 2 x}} - 1 \right)}}{2} + \frac{\log{\left(\sqrt{1 - e^{- 2 x}} + 1 \right)}}{2} 
log(1 + sqrt(1 - exp(-2*x)))/2 - log(-1 + sqrt(1 - exp(-2*x)))/2
Denominador racional [src] 
     /        ___________\      /       ___________\
     |       /      -2*x |      |      /      -2*x |
- log\-1 + \/  1 - e     / + log\1 + \/  1 - e     /
----------------------------------------------------
                         2
−log⁡(1−e−2x−1)+log⁡(1−e−2x+1)2\frac{- \log{\left(\sqrt{1 - e^{- 2 x}} - 1 \right)} + \log{\left(\sqrt{1 - e^{- 2 x}} + 1 \right)}}{2} 
(-log(-1 + sqrt(1 - exp(-2*x))) + log(1 + sqrt(1 - exp(-2*x))))/2
Unión de expresiones racionales [src] 
     /        ___________________\      /       ___________________\
     |       / /      2*x\  -2*x |      |      / /      2*x\  -2*x |
- log\-1 + \/  \-1 + e   /*e     / + log\1 + \/  \-1 + e   /*e     /
--------------------------------------------------------------------
                                 2
−log⁡((e2x−1)e−2x−1)+log⁡((e2x−1)e−2x+1)2\frac{- \log{\left(\sqrt{\left(e^{2 x} - 1\right) e^{- 2 x}} - 1 \right)} + \log{\left(\sqrt{\left(e^{2 x} - 1\right) e^{- 2 x}} + 1 \right)}}{2} 
(-log(-1 + sqrt((-1 + exp(2*x))*exp(-2*x))) + log(1 + sqrt((-1 + exp(2*x))*exp(-2*x))))/2
Potencias [src] 
   /       ___________\      /        ___________\
   |      /      -2*x |      |       /      -2*x |
log\1 + \/  1 - e     /   log\-1 + \/  1 - e     /
----------------------- - ------------------------
           2                         2
−log⁡(1−e−2x−1)2+log⁡(1−e−2x+1)2- \frac{\log{\left(\sqrt{1 - e^{- 2 x}} - 1 \right)}}{2} + \frac{\log{\left(\sqrt{1 - e^{- 2 x}} + 1 \right)}}{2} 
log(1 + sqrt(1 - exp(-2*x)))/2 - log(-1 + sqrt(1 - exp(-2*x)))/2
Compilar la expresión [src] 
   /       ___________\      /        ___________\
   |      /      -2*x |      |       /      -2*x |
log\1 + \/  1 - e     /   log\-1 + \/  1 - e     /
----------------------- - ------------------------
           2                         2
−log⁡(1−e−2x−1)2+log⁡(1−e−2x+1)2- \frac{\log{\left(\sqrt{1 - e^{- 2 x}} - 1 \right)}}{2} + \frac{\log{\left(\sqrt{1 - e^{- 2 x}} + 1 \right)}}{2} 
log(1 + sqrt(1 - exp(-2*x)))/2 - log(-1 + sqrt(1 - exp(-2*x)))/2
Parte trigonométrica [src] 
   /       _____________________________\      /        _____________________________\
   |      /                        -2*x |      |       /                        -2*x |
log\1 + \/  1 - (cosh(1) + sinh(1))     /   log\-1 + \/  1 - (cosh(1) + sinh(1))     /
----------------------------------------- - ------------------------------------------
         2*log(cosh(1) + sinh(1))                    2*log(cosh(1) + sinh(1))
−log⁡(1−(sinh⁡(1)+cosh⁡(1))−2x−1)2log⁡(sinh⁡(1)+cosh⁡(1))+log⁡(1−(sinh⁡(1)+cosh⁡(1))−2x+1)2log⁡(sinh⁡(1)+cosh⁡(1))- \frac{\log{\left(\sqrt{1 - \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{- 2 x}} - 1 \right)}}{2 \log{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)} \right)}} + \frac{\log{\left(\sqrt{1 - \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{- 2 x}} + 1 \right)}}{2 \log{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)} \right)}} 
   /      ___________________________\      /       ___________________________\
log\1 + \/ 1 - cosh(2*x) + sinh(2*x) /   log\-1 + \/ 1 - cosh(2*x) + sinh(2*x) /
-------------------------------------- - ---------------------------------------
                  2                                         2
−log⁡(sinh⁡(2x)−cosh⁡(2x)+1−1)2+log⁡(sinh⁡(2x)−cosh⁡(2x)+1+1)2- \frac{\log{\left(\sqrt{\sinh{\left(2 x \right)} - \cosh{\left(2 x \right)} + 1} - 1 \right)}}{2} + \frac{\log{\left(\sqrt{\sinh{\left(2 x \right)} - \cosh{\left(2 x \right)} + 1} + 1 \right)}}{2} 
   /       ___________\      /        ___________\
   |      /      -2*x |      |       /      -2*x |
log\1 + \/  1 - e     /   log\-1 + \/  1 - e     /
----------------------- - ------------------------
           2                         2
−log⁡(1−e−2x−1)2+log⁡(1−e−2x+1)2- \frac{\log{\left(\sqrt{1 - e^{- 2 x}} - 1 \right)}}{2} + \frac{\log{\left(\sqrt{1 - e^{- 2 x}} + 1 \right)}}{2} 
 /     /       _____________________________\      /        _____________________________\\ 
 |     |      /                        -2*x |      |       /                        -2*x || 
-\- log\1 + \/  1 - (cosh(1) + sinh(1))     / + log\-1 + \/  1 - (cosh(1) + sinh(1))     // 
--------------------------------------------------------------------------------------------
                                  2*log(cosh(1) + sinh(1))
−log⁡(1−(sinh⁡(1)+cosh⁡(1))−2x−1)−log⁡(1−(sinh⁡(1)+cosh⁡(1))−2x+1)2log⁡(sinh⁡(1)+cosh⁡(1))- \frac{\log{\left(\sqrt{1 - \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{- 2 x}} - 1 \right)} - \log{\left(\sqrt{1 - \left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{- 2 x}} + 1 \right)}}{2 \log{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)} \right)}} 
-(-log(1 + sqrt(1 - (cosh(1) + sinh(1))^(-2*x))) + log(-1 + sqrt(1 - (cosh(1) + sinh(1))^(-2*x))))/(2*log(cosh(1) + sinh(1)))
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