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Simplificación de expresiones/ Descomposición en fracciones simples/ Piecewise((-acosh(1/x)+1/(x*sqrt(-1+x^(-2)))-x/sqrt(-1+x^(-2)),1/|x^2|>1),(i*asin(1/x)+i*x/sqrt(1-1/x^2)-i/(x*sqrt(1-1/x^2)),True))

¿Cómo vas a descomponer esta Piecewise((-acosh(1/x)+1/(x*sqrt(-1+x^(-2)))-x/sqrt(-1+x^(-2)),1/|x^2|>1),(i*asin(1/x)+i*x/sqrt(1-1/x^2)-i/(x*sqrt(1-1/x^2)),True)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
/       /1\          1                 x              1      
|- acosh|-| + ---------------- - --------------  for ---- > 1
|       \x/          _________        _________      | 2|    
|                   /      1         /      1        |x |    
|             x*   /  -1 + --       /  -1 + --               
|                 /         2      /         2               
|               \/         x     \/         x                
<                                                            
|       /1\        I*x               I                       
| I*asin|-| + ------------- - ---------------     otherwise  
|       \x/        ________          ________                
|                 /     1           /     1                  
|                /  1 - --    x*   /  1 - --                 
|               /        2        /        2                 
\             \/        x       \/        x
{−x−1+1x2−acosh⁡(1x)+1x−1+1x2for 1∣x2∣>1ix1−1x2+iasin⁡(1x)−ix1−1x2otherwise\begin{cases} - \frac{x}{\sqrt{-1 + \frac{1}{x^{2}}}} - \operatorname{acosh}{\left(\frac{1}{x} \right)} + \frac{1}{x \sqrt{-1 + \frac{1}{x^{2}}}} & \text{for}\: \frac{1}{\left|{x^{2}}\right|} > 1 \\\frac{i x}{\sqrt{1 - \frac{1}{x^{2}}}} + i \operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{i}{x \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases} 
Piecewise((-acosh(1/x) + 1/(x*sqrt(-1 + x^(-2))) - x/sqrt(-1 + x^(-2)), 1/|x^2| > 1), (i*asin(1/x) + i*x/sqrt(1 - 1/x^2) - i/(x*sqrt(1 - 1/x^2)), True))
Simplificación general [src] 
/                     _________               
|        /1\         /      1          1      
| - acosh|-| + x*   /  -1 + --    for ---- > 1
|        \x/       /         2        | 2|    
|                \/         x         |x |    
|                                             
<  /        _________          \              
|  |       /       2           |              
|  |      /  -1 + x         /1\|              
|I*|x*   /   -------  + asin|-||   otherwise  
|  |    /        2          \x/|              
|  \  \/        x              /              
\
{x−1+1x2−acosh⁡(1x)for 1∣x2∣>1i(xx2−1x2+asin⁡(1x))otherwise\begin{cases} x \sqrt{-1 + \frac{1}{x^{2}}} - \operatorname{acosh}{\left(\frac{1}{x} \right)} & \text{for}\: \frac{1}{\left|{x^{2}}\right|} > 1 \\i \left(x \sqrt{\frac{x^{2} - 1}{x^{2}}} + \operatorname{asin}{\left(\frac{1}{x} \right)}\right) & \text{otherwise} \end{cases} 
Piecewise((-acosh(1/x) + x*sqrt(-1 + x^(-2)), 1/|x^2| > 1), (i*(x*sqrt((-1 + x^2)/x^2) + asin(1/x)), True))
Unión de expresiones racionales [src] 
/                   ________                          
|                  /      2                           
|       2         /  1 - x        /1\                 
|  1 - x  - x*   /   ------ *acosh|-|                 
|               /       2         \x/                 
|             \/       x                       1      
|  ----------------------------------     for ---- > 1
|                   ________                  | 2|    
|                  /      2                   |x |    
|                 /  1 - x                            
|           x*   /   ------                           
|               /       2                             
|             \/       x                              
<                                                     
|  /                  _________        \              
|  |                 /       2         |              
|  |      2         /  -1 + x       /1\|              
|I*|-1 + x  + x*   /   ------- *asin|-||              
|  |              /        2        \x/|              
|  \            \/        x            /              
|---------------------------------------   otherwise  
|                   _________                         
|                  /       2                          
|                 /  -1 + x                           
|           x*   /   -------                          
|               /        2                            
\             \/        x
{−x2−x1−x2x2acosh⁡(1x)+1x1−x2x2for 1∣x2∣>1i(x2+xx2−1x2asin⁡(1x)−1)xx2−1x2otherwise\begin{cases} \frac{- x^{2} - x \sqrt{\frac{1 - x^{2}}{x^{2}}} \operatorname{acosh}{\left(\frac{1}{x} \right)} + 1}{x \sqrt{\frac{1 - x^{2}}{x^{2}}}} & \text{for}\: \frac{1}{\left|{x^{2}}\right|} > 1 \\\frac{i \left(x^{2} + x \sqrt{\frac{x^{2} - 1}{x^{2}}} \operatorname{asin}{\left(\frac{1}{x} \right)} - 1\right)}{x \sqrt{\frac{x^{2} - 1}{x^{2}}}} & \text{otherwise} \end{cases} 
Piecewise(((1 - x^2 - x*sqrt((1 - x^2)/x^2)*acosh(1/x))/(x*sqrt((1 - x^2)/x^2)), 1/|x^2| > 1), (i*(-1 + x^2 + x*sqrt((-1 + x^2)/x^2)*asin(1/x))/(x*sqrt((-1 + x^2)/x^2)), True))
Respuesta numérica [src] 
Piecewise((-acosh(1/x) + (-1.0 + x^(-2))^(-0.5)/x - x*(-1.0 + x^(-2))^(-0.5), 1/|x^2| > 1), (i*asin(1/x) + i*x*(1.0 - 1/x^2)^(-0.5) - i*(1.0 - 1/x^2)^(-0.5)/x, True))
Piecewise((-acosh(1/x) + (-1.0 + x^(-2))^(-0.5)/x - x*(-1.0 + x^(-2))^(-0.5), 1/|x^2| > 1), (i*asin(1/x) + i*x*(1.0 - 1/x^2)^(-0.5) - i*(1.0 - 1/x^2)^(-0.5)/x, True))
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