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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   /      ________    \
   |     /      2     |
   |    /      a      |
   |   /   1 - --  - 1|
   |  /         2     |
   |\/         x      |
log|------------------|
   |      ________    |
   |     /      2     |
   |    /      a      |
   |   /   1 - --  + 1|
   |  /         2     |
   \\/         x      /
log⁡(−a2x2+1−1−a2x2+1+1)\log{\left(\frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1} - 1}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} \right)} 
log((sqrt(1 - a^2/x^2) - 1)/(sqrt(1 - a^2/x^2) + 1))
Descomposición de una fracción [src] 
log(-1/(1 + sqrt(1 - a^2/x^2)) + sqrt(1 - a^2/x^2)/(1 + sqrt(1 - a^2/x^2)))
log⁡(−a2x2+1−a2x2+1+1−1−a2x2+1+1)\log{\left(\frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} - \frac{1}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} \right)} 
   /                               ________  \
   |                              /      2   |
   |                             /      a    |
   |                            /   1 - --   |
   |                           /         2   |
   |          1              \/         x    |
log|- ------------------ + ------------------|
   |            ________             ________|
   |           /      2             /      2 |
   |          /      a             /      a  |
   |  1 +    /   1 - --    1 +    /   1 - -- |
   |        /         2          /         2 |
   \      \/         x         \/         x  /
Respuesta numérica [src] 
log((sqrt(1 - a^2/x^2) - 1)/(sqrt(1 - a^2/x^2) + 1))
log((sqrt(1 - a^2/x^2) - 1)/(sqrt(1 - a^2/x^2) + 1))
Denominador común [src] 
   /                               ________  \
   |                              /      2   |
   |                             /      a    |
   |                            /   1 - --   |
   |                           /         2   |
   |          1              \/         x    |
log|- ------------------ + ------------------|
   |            ________             ________|
   |           /      2             /      2 |
   |          /      a             /      a  |
   |  1 +    /   1 - --    1 +    /   1 - -- |
   |        /         2          /         2 |
   \      \/         x         \/         x  /
log⁡(−a2x2+1−a2x2+1+1−1−a2x2+1+1)\log{\left(\frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} - \frac{1}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} \right)} 
log(-1/(1 + sqrt(1 - a^2/x^2)) + sqrt(1 - a^2/x^2)/(1 + sqrt(1 - a^2/x^2)))
Unión de expresiones racionales [src] 
   /           _________\
   |          /  2    2 |
   |         /  x  - a  |
   |-1 +    /   ------- |
   |       /        2   |
   |     \/        x    |
log|--------------------|
   |          _________ |
   |         /  2    2  |
   |        /  x  - a   |
   |1 +    /   -------  |
   |      /        2    |
   \    \/        x     /
log⁡(−a2+x2x2−1−a2+x2x2+1)\log{\left(\frac{\sqrt{\frac{- a^{2} + x^{2}}{x^{2}}} - 1}{\sqrt{\frac{- a^{2} + x^{2}}{x^{2}}} + 1} \right)} 
log((-1 + sqrt((x^2 - a^2)/x^2))/(1 + sqrt((x^2 - a^2)/x^2)))
Combinatoria [src] 
   /                               ________  \
   |                              /      2   |
   |                             /      a    |
   |                            /   1 - --   |
   |                           /         2   |
   |          1              \/         x    |
log|- ------------------ + ------------------|
   |            ________             ________|
   |           /      2             /      2 |
   |          /      a             /      a  |
   |  1 +    /   1 - --    1 +    /   1 - -- |
   |        /         2          /         2 |
   \      \/         x         \/         x  /
log⁡(−a2x2+1−a2x2+1+1−1−a2x2+1+1)\log{\left(\frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} - \frac{1}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} \right)} 
log(-1/(1 + sqrt(1 - a^2/x^2)) + sqrt(1 - a^2/x^2)/(1 + sqrt(1 - a^2/x^2)))
Denominador racional [src] 
   /    /          ________         ________         ________       ________\ \
   |    |         /      2         /      2         /      2       /      2 | |
   |  2 |        /      a         /      a         /      a       /      a  | |
   |-x *|1 -    /   1 - --  -    /   1 - --  +    /   1 - -- *   /   1 - -- | |
   |    |      /         2      /         2      /         2    /         2 | |
   |    \    \/         x     \/         x     \/         x   \/         x  / |
log|--------------------------------------------------------------------------|
   |                                     2                                    |
   \                                    a                                     /
log⁡(−x2(−a2x2+1−a2x2+1−−a2x2+1−−a2x2+1+1)a2)\log{\left(- \frac{x^{2} \left(\sqrt{- \frac{a^{2}}{x^{2}} + 1} \sqrt{- \frac{a^{2}}{x^{2}} + 1} - \sqrt{- \frac{a^{2}}{x^{2}} + 1} - \sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1\right)}{a^{2}} \right)} 
log(-x^2*(1 - sqrt(1 - a^2/x^2) - sqrt(1 - a^2/x^2) + sqrt(1 - a^2/x^2)*sqrt(1 - a^2/x^2))/a^2)
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