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Simplificación de expresiones/ Descomposición en fracciones simples/ Piecewise((-a^2*acosh(x/a)/2+x^3/(2*a*sqrt(-1+x^2/a^2))-a*x/(2*sqrt(-1+x^2/a^2)),|x^2/a^2|>1),(i*a^2*asin(x/a)/2+i*a*x*sqrt(1-x^2/a^2)/2,True))

¿Cómo vas a descomponer esta Piecewise((-a^2*acosh(x/a)/2+x^3/(2*a*sqrt(-1+x^2/a^2))-a*x/(2*sqrt(-1+x^2/a^2)),|x^2/a^2|>1),(i*a^2*asin(x/a)/2+i*a*x*sqrt(1-x^2/a^2)/2,True)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
/   2      /x\                                                        
|  a *acosh|-|             3                                  | 2|    
|          \a/            x                   a*x             |x |    
|- ----------- + ------------------- - -----------------  for |--| > 1
|       2                  _________           _________      | 2|    
|                         /       2           /       2       |a |    
|                        /       x           /       x                
|                2*a*   /   -1 + --    2*   /   -1 + --               
|                      /          2        /          2               
<                    \/          a       \/          a                
|                                                                     
|                                     ________                        
|                                    /      2                         
|                                   /      x                          
|             2     /x\   I*a*x*   /   1 - --                         
|          I*a *asin|-|           /         2                         
|                   \a/         \/         a                          
|          ------------ + --------------------             otherwise  
\               2                  2
{−a2acosh⁡(xa)2−ax2−1+x2a2+x32a−1+x2a2for ∣x2a2∣>1ia2asin⁡(xa)2+iax1−x2a22otherwise\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{2} - \frac{a x}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{x^{3}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{i a x \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases} 
Piecewise((-a^2*acosh(x/a)/2 + x^3/(2*a*sqrt(-1 + x^2/a^2)) - a*x/(2*sqrt(-1 + x^2/a^2)), |x^2/a^2| > 1), (i*a^2*asin(x/a)/2 + i*a*x*sqrt(1 - x^2/a^2)/2, True))
Simplificación general [src] 
/  /        _________             \               
|  |       /  2    2              |               
|  |      /  x  - a            /x\|               
|a*|x*   /   -------  - a*acosh|-||               
|  |    /        2             \a/|       | 2|    
|  \  \/        a                 /       |x |    
|----------------------------------   for |--| > 1
|                2                        | 2|    
|                                         |a |    
<                                                 
|    /                    _________\              
|    |                   /  2    2 |              
|    |      /x\         /  a  - x  |              
|I*a*|a*asin|-| + x*   /   ------- |              
|    |      \a/       /        2   |              
|    \              \/        a    /              
|-----------------------------------   otherwise  
|                 2                               
\
{a(−aacosh⁡(xa)+x−a2+x2a2)2for ∣x2a2∣>1ia(aasin⁡(xa)+xa2−x2a2)2otherwise\begin{cases} \frac{a \left(- a \operatorname{acosh}{\left(\frac{x}{a} \right)} + x \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}}\right)}{2} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{i a \left(a \operatorname{asin}{\left(\frac{x}{a} \right)} + x \sqrt{\frac{a^{2} - x^{2}}{a^{2}}}\right)}{2} & \text{otherwise} \end{cases} 
Piecewise((a*(x*sqrt((x^2 - a^2)/a^2) - a*acosh(x/a))/2, |x^2/a^2| > 1), (i*a*(a*asin(x/a) + x*sqrt((a^2 - x^2)/a^2))/2, True))
Respuesta numérica [src] 
Piecewise((-0.5*a^2*acosh(x/a) + 0.5*x^3*(-1.0 + x^2/a^2)^(-0.5)/a - 0.5*a*x*(-1.0 + x^2/a^2)^(-0.5), |x^2/a^2| > 1), (0.5*i*a^2*asin(x/a) + 0.5*i*a*x*(1.0 - x^2/a^2)^0.5, True))
Piecewise((-0.5*a^2*acosh(x/a) + 0.5*x^3*(-1.0 + x^2/a^2)^(-0.5)/a - 0.5*a*x*(-1.0 + x^2/a^2)^(-0.5), |x^2/a^2| > 1), (0.5*i*a^2*asin(x/a) + 0.5*i*a*x*(1.0 - x^2/a^2)^0.5, True))
Unión de expresiones racionales [src] 
/                     _________                       
|                    /  2    2                        
| 3      2    3     /  x  - a        /x\              
|x  - x*a  - a *   /   ------- *acosh|-|              
|                 /        2         \a/      | 2|    
|               \/        a                   |x |    
|---------------------------------------  for |--| > 1
|                    _________                | 2|    
|                   /  2    2                 |a |    
|                  /  x  - a                          
|          2*a*   /   -------                         
<                /        2                           
|              \/        a                            
|                                                     
|      /                    _________\                
|      |                   /  2    2 |                
|      |      /x\         /  a  - x  |                
|  I*a*|a*asin|-| + x*   /   ------- |                
|      |      \a/       /        2   |                
|      \              \/        a    /                
|  -----------------------------------     otherwise  
|                   2                                 
\
{−a3−a2+x2a2acosh⁡(xa)−a2x+x32a−a2+x2a2for ∣x2a2∣>1ia(aasin⁡(xa)+xa2−x2a2)2otherwise\begin{cases} \frac{- a^{3} \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}} \operatorname{acosh}{\left(\frac{x}{a} \right)} - a^{2} x + x^{3}}{2 a \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{i a \left(a \operatorname{asin}{\left(\frac{x}{a} \right)} + x \sqrt{\frac{a^{2} - x^{2}}{a^{2}}}\right)}{2} & \text{otherwise} \end{cases} 
Piecewise(((x^3 - x*a^2 - a^3*sqrt((x^2 - a^2)/a^2)*acosh(x/a))/(2*a*sqrt((x^2 - a^2)/a^2)), |x^2/a^2| > 1), (i*a*(a*asin(x/a) + x*sqrt((a^2 - x^2)/a^2))/2, True))
Potencias [src] 
/   2      /x\                                                        
|  a *acosh|-|             3                                  | 2|    
|          \a/            x                   a*x             |x |    
|- ----------- + ------------------- - -----------------  for |--| > 1
|       2                  _________           _________      | 2|    
|                         /       2           /       2       |a |    
|                        /       x           /       x                
|                2*a*   /   -1 + --    2*   /   -1 + --               
|                      /          2        /          2               
<                    \/          a       \/          a                
|                                                                     
|                                   _________                         
|                                  /       2                          
|                                 /       x                           
|             2     /x\   a*x*   /   -1 + --                          
|          I*a *asin|-|         /          2                          
|                   \a/       \/          a                           
|          ------------ + -------------------              otherwise  
\               2                  2
{−a2acosh⁡(xa)2−ax2−1+x2a2+x32a−1+x2a2for ∣x2a2∣>1ia2asin⁡(xa)2+ax−1+x2a22otherwise\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{2} - \frac{a x}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{x^{3}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{a x \sqrt{-1 + \frac{x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases} 
Piecewise((-a^2*acosh(x/a)/2 + x^3/(2*a*sqrt(-1 + x^2/a^2)) - a*x/(2*sqrt(-1 + x^2/a^2)), |x^2/a^2| > 1), (i*a^2*asin(x/a)/2 + a*x*sqrt(-1 + x^2/a^2)/2, True))
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