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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
/                                          3                   
|    I*acosh(x)        I*x              I*x            | 2|    
|x + ---------- + -------------- - --------------  for |x | > 1
|        2             _________        _________              
|                     /       2        /       2               
|                 2*\/  -1 + x     2*\/  -1 + x                
<                                                              
|                             ________                         
|                            /      2                          
|              asin(x)   x*\/  1 - x                           
|          x - ------- - -------------              otherwise  
|                 2            2                               
\
{−ix32x2−1+x+ix2x2−1+iacosh⁡(x)2for ∣x2∣>1−x1−x22+x−asin⁡(x)2otherwise\begin{cases} - \frac{i x^{3}}{2 \sqrt{x^{2} - 1}} + x + \frac{i x}{2 \sqrt{x^{2} - 1}} + \frac{i \operatorname{acosh}{\left(x \right)}}{2} & \text{for}\: \left|{x^{2}}\right| > 1 \\- \frac{x \sqrt{1 - x^{2}}}{2} + x - \frac{\operatorname{asin}{\left(x \right)}}{2} & \text{otherwise} \end{cases} 
Piecewise((x + i*acosh(x)/2 + i*x/(2*sqrt(-1 + x^2)) - i*x^3/(2*sqrt(-1 + x^2)), |x^2| > 1), (x - asin(x)/2 - x*sqrt(1 - x^2)/2, True))
Simplificación general [src] 
/         _________                                        
|        /       2                          3              
|I*x + \/  -1 + x  *(2*x + I*acosh(x)) - I*x       | 2|    
|--------------------------------------------  for |x | > 1
|                    _________                             
|                   /       2                              
<               2*\/  -1 + x                               
|                                                          
|                           ________                       
|                          /      2                        
|            asin(x)   x*\/  1 - x                         
|        x - ------- - -------------            otherwise  
\               2            2
{−ix3+ix+(2x+iacosh⁡(x))x2−12x2−1for ∣x2∣>1−x1−x22+x−asin⁡(x)2otherwise\begin{cases} \frac{- i x^{3} + i x + \left(2 x + i \operatorname{acosh}{\left(x \right)}\right) \sqrt{x^{2} - 1}}{2 \sqrt{x^{2} - 1}} & \text{for}\: \left|{x^{2}}\right| > 1 \\- \frac{x \sqrt{1 - x^{2}}}{2} + x - \frac{\operatorname{asin}{\left(x \right)}}{2} & \text{otherwise} \end{cases} 
Piecewise(((i*x + sqrt(-1 + x^2)*(2*x + i*acosh(x)) - i*x^3)/(2*sqrt(-1 + x^2)), |x^2| > 1), (x - asin(x)/2 - x*sqrt(1 - x^2)/2, True))
Respuesta numérica [src] 
Piecewise((x + 0.5*i*acosh(x) + 0.5*i*x*(-1.0 + x^2)^(-0.5) - 0.5*i*x^3*(-1.0 + x^2)^(-0.5), |x^2| > 1), (x - 0.5*asin(x) - 0.5*x*(1.0 - x^2)^0.5, True))
Piecewise((x + 0.5*i*acosh(x) + 0.5*i*x*(-1.0 + x^2)^(-0.5) - 0.5*i*x^3*(-1.0 + x^2)^(-0.5), |x^2| > 1), (x - 0.5*asin(x) - 0.5*x*(1.0 - x^2)^0.5, True))
Unión de expresiones racionales [src] 
/                    _________        _________                       
|         3         /       2        /       2                        
|I*x - I*x  + 2*x*\/  -1 + x   + I*\/  -1 + x  *acosh(x)      | 2|    
|-------------------------------------------------------  for |x | > 1
|                          _________                                  
|                         /       2                                   
<                     2*\/  -1 + x                                    
|                                                                     
|                                  ________                           
|                                 /      2                            
|            -asin(x) + 2*x - x*\/  1 - x                             
|            ------------------------------                otherwise  
\                          2
{−ix3+2xx2−1+ix+ix2−1acosh⁡(x)2x2−1for ∣x2∣>1−x1−x2+2x−asin⁡(x)2otherwise\begin{cases} \frac{- i x^{3} + 2 x \sqrt{x^{2} - 1} + i x + i \sqrt{x^{2} - 1} \operatorname{acosh}{\left(x \right)}}{2 \sqrt{x^{2} - 1}} & \text{for}\: \left|{x^{2}}\right| > 1 \\\frac{- x \sqrt{1 - x^{2}} + 2 x - \operatorname{asin}{\left(x \right)}}{2} & \text{otherwise} \end{cases} 
Piecewise(((i*x - i*x^3 + 2*x*sqrt(-1 + x^2) + i*sqrt(-1 + x^2)*acosh(x))/(2*sqrt(-1 + x^2)), |x^2| > 1), ((-asin(x) + 2*x - x*sqrt(1 - x^2))/2, True))
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