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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
/           /    ___\                                                                
|           |x*\/ b |                                                                
|  I*a*acosh|-------|                                                                
|           |   ___ |             ___                      3               |   2|    
|           \ \/ a  /       I*x*\/ a                  I*b*x                |b*x |    
|- ------------------ - ------------------ + ------------------------  for |----| > 1
|           ___                ___________                ___________      | a  |    
|       2*\/ b                /         2                /         2                 
|                            /       b*x         ___    /       b*x                  
|                       2*  /   -1 + ----    2*\/ a *  /   -1 + ----                 
<                         \/          a              \/          a                   
|                                                                                    
|                   /    ___\                __________                              
|                   |x*\/ b |               /        2                               
|             a*asin|-------|       ___    /      b*x                                
|                   |   ___ |   x*\/ a *  /   1 - ----                               
|                   \ \/ a  /           \/         a                                 
|             --------------- + -----------------------                  otherwise   
|                     ___                  2                                         
|                 2*\/ b                                                             
\
{−iax2−1+bx2a−iaacosh⁡(bxa)2b+ibx32a−1+bx2afor ∣bx2a∣>1ax1−bx2a2+aasin⁡(bxa)2botherwise\begin{cases} - \frac{i \sqrt{a} x}{2 \sqrt{-1 + \frac{b x^{2}}{a}}} - \frac{i a \operatorname{acosh}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)}}{2 \sqrt{b}} + \frac{i b x^{3}}{2 \sqrt{a} \sqrt{-1 + \frac{b x^{2}}{a}}} & \text{for}\: \left|{\frac{b x^{2}}{a}}\right| > 1 \\\frac{\sqrt{a} x \sqrt{1 - \frac{b x^{2}}{a}}}{2} + \frac{a \operatorname{asin}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)}}{2 \sqrt{b}} & \text{otherwise} \end{cases} 
Piecewise((-i*a*acosh(x*sqrt(b)/sqrt(a))/(2*sqrt(b)) - i*x*sqrt(a)/(2*sqrt(-1 + b*x^2/a)) + i*b*x^3/(2*sqrt(a)*sqrt(-1 + b*x^2/a)), |b*x^2/a| > 1), (a*asin(x*sqrt(b)/sqrt(a))/(2*sqrt(b)) + x*sqrt(a)*sqrt(1 - b*x^2/a)/2, True))
Simplificación general [src] 
/        /             ___________                                                     ___________\                
|        |            /         2                           /    ___\                 /         2 |                
|    ___ | 3/2  3    /  -a + b*x       ___ /        2\      |x*\/ b |         ___    /  -a + b*x  |                
|I*\/ a *|b   *x *  /   ---------  - \/ a *\-a + b*x /*acosh|-------| - a*x*\/ b *  /   --------- |                
|        |        \/        a                               |   ___ |             \/        a     |      |   2|    
|        \                                                  \ \/ a  /                             /      |b*x |    
|--------------------------------------------------------------------------------------------------  for |----| > 1
|                                           ___ /        2\                                              | a  |    
|                                       2*\/ b *\-a + b*x /                                                        
<                                                                                                                  
|                                  /    ___\                __________                                             
|                                  |x*\/ b |               /        2                                              
|                            a*asin|-------|       ___    /      b*x                                               
|                                  |   ___ |   x*\/ a *  /   1 - ----                                              
|                                  \ \/ a  /           \/         a                                                
|                            --------------- + -----------------------                                 otherwise   
|                                    ___                  2                                                        
|                                2*\/ b                                                                            
\
{ia(−a(−a+bx2)acosh⁡(bxa)−abx−a+bx2a+b32x3−a+bx2a)2b(−a+bx2)for ∣bx2a∣>1ax1−bx2a2+aasin⁡(bxa)2botherwise\begin{cases} \frac{i \sqrt{a} \left(- \sqrt{a} \left(- a + b x^{2}\right) \operatorname{acosh}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)} - a \sqrt{b} x \sqrt{\frac{- a + b x^{2}}{a}} + b^{\frac{3}{2}} x^{3} \sqrt{\frac{- a + b x^{2}}{a}}\right)}{2 \sqrt{b} \left(- a + b x^{2}\right)} & \text{for}\: \left|{\frac{b x^{2}}{a}}\right| > 1 \\\frac{\sqrt{a} x \sqrt{1 - \frac{b x^{2}}{a}}}{2} + \frac{a \operatorname{asin}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)}}{2 \sqrt{b}} & \text{otherwise} \end{cases} 
Piecewise((i*sqrt(a)*(b^(3/2)*x^3*sqrt((-a + b*x^2)/a) - sqrt(a)*(-a + b*x^2)*acosh(x*sqrt(b)/sqrt(a)) - a*x*sqrt(b)*sqrt((-a + b*x^2)/a))/(2*sqrt(b)*(-a + b*x^2)), |b*x^2/a| > 1), (a*asin(x*sqrt(b)/sqrt(a))/(2*sqrt(b)) + x*sqrt(a)*sqrt(1 - b*x^2/a)/2, True))
Respuesta numérica [src] 
Piecewise((-0.5*i*a*b^(-0.5)*acosh(x*sqrt(b)/sqrt(a)) - 0.5*i*x*a^0.5*(-1.0 + b*x^2/a)^(-0.5) + 0.5*i*b*a^(-0.5)*x^3*(-1.0 + b*x^2/a)^(-0.5), |b*x^2/a| > 1), (0.5*a*b^(-0.5)*asin(x*sqrt(b)/sqrt(a)) + 0.5*x*a^0.5*(1.0 - b*x^2/a)^0.5, True))
Piecewise((-0.5*i*a*b^(-0.5)*acosh(x*sqrt(b)/sqrt(a)) - 0.5*i*x*a^0.5*(-1.0 + b*x^2/a)^(-0.5) + 0.5*i*b*a^(-0.5)*x^3*(-1.0 + b*x^2/a)^(-0.5), |b*x^2/a| > 1), (0.5*a*b^(-0.5)*asin(x*sqrt(b)/sqrt(a)) + 0.5*x*a^0.5*(1.0 - b*x^2/a)^0.5, True))
Potencias [src] 
/                                                         /    ___\                
|                                                         |x*\/ b |                
|                                                I*a*acosh|-------|                
|           ____                     3                    |   ___ |      |   2|    
|       x*\/ -a                 I*b*x                     \ \/ a  /      |b*x |    
|- ------------------ + ---------------------- - ------------------  for |----| > 1
|         ___________          _______________            ___            | a  |    
|        /         2          /   /        2\         2*\/ b                       
|       /       b*x          /    |     b*x |                                      
|  2*  /   -1 + ----    2*  /   a*|-1 + ----|                                      
<    \/          a        \/      \      a  /                                      
|                                                                                  
|                    ______________         /    ___\                              
|                   /   /       2\          |x*\/ b |                              
|                  /    |    b*x |    a*asin|-------|                              
|             x*  /   a*|1 - ----|          |   ___ |                              
|               \/      \     a  /          \ \/ a  /                              
|             --------------------- + ---------------                  otherwise   
|                       2                     ___                                  
|                                         2*\/ b                                   
\
{−iaacosh⁡(bxa)2b+ibx32a(−1+bx2a)−x−a2−1+bx2afor ∣bx2a∣>1aasin⁡(bxa)2b+xa(1−bx2a)2otherwise\begin{cases} - \frac{i a \operatorname{acosh}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)}}{2 \sqrt{b}} + \frac{i b x^{3}}{2 \sqrt{a \left(-1 + \frac{b x^{2}}{a}\right)}} - \frac{x \sqrt{- a}}{2 \sqrt{-1 + \frac{b x^{2}}{a}}} & \text{for}\: \left|{\frac{b x^{2}}{a}}\right| > 1 \\\frac{a \operatorname{asin}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)}}{2 \sqrt{b}} + \frac{x \sqrt{a \left(1 - \frac{b x^{2}}{a}\right)}}{2} & \text{otherwise} \end{cases} 
Piecewise((-x*sqrt(-a)/(2*sqrt(-1 + b*x^2/a)) + i*b*x^3/(2*sqrt(a*(-1 + b*x^2/a))) - i*a*acosh(x*sqrt(b)/sqrt(a))/(2*sqrt(b)), |b*x^2/a| > 1), (x*sqrt(a*(1 - b*x^2/a))/2 + a*asin(x*sqrt(b)/sqrt(a))/(2*sqrt(b)), True))
Unión de expresiones racionales [src] 
/  /                                ___________               \                
|  |                               /         2       /    ___\|                
|  | 3/2  3         ___    3/2    /  -a + b*x        |x*\/ b ||                
|I*|b   *x  - a*x*\/ b  - a   *  /   --------- *acosh|-------||                
|  |                           \/        a           |   ___ ||      |   2|    
|  \                                                 \ \/ a  //      |b*x |    
|--------------------------------------------------------------  for |----| > 1
|                                   ___________                      | a  |    
|                                  /         2                                 
|                    ___   ___    /  -a + b*x                                  
|                2*\/ a *\/ b *  /   ---------                                 
<                              \/        a                                     
|                                                                              
|                                            __________                        
|             /    ___\                     /        2                         
|             |x*\/ b |       ___   ___    /  a - b*x                          
|       a*asin|-------| + x*\/ a *\/ b *  /   --------                         
|             |   ___ |                 \/       a                             
|             \ \/ a  /                                                        
|       -----------------------------------------------            otherwise   
|                               ___                                            
|                           2*\/ b                                             
\
{i(−a32−a+bx2aacosh⁡(bxa)−abx+b32x3)2ab−a+bx2afor ∣bx2a∣>1abxa−bx2a+aasin⁡(bxa)2botherwise\begin{cases} \frac{i \left(- a^{\frac{3}{2}} \sqrt{\frac{- a + b x^{2}}{a}} \operatorname{acosh}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)} - a \sqrt{b} x + b^{\frac{3}{2}} x^{3}\right)}{2 \sqrt{a} \sqrt{b} \sqrt{\frac{- a + b x^{2}}{a}}} & \text{for}\: \left|{\frac{b x^{2}}{a}}\right| > 1 \\\frac{\sqrt{a} \sqrt{b} x \sqrt{\frac{a - b x^{2}}{a}} + a \operatorname{asin}{\left(\frac{\sqrt{b} x}{\sqrt{a}} \right)}}{2 \sqrt{b}} & \text{otherwise} \end{cases} 
Piecewise((i*(b^(3/2)*x^3 - a*x*sqrt(b) - a^(3/2)*sqrt((-a + b*x^2)/a)*acosh(x*sqrt(b)/sqrt(a)))/(2*sqrt(a)*sqrt(b)*sqrt((-a + b*x^2)/a)), |b*x^2/a| > 1), ((a*asin(x*sqrt(b)/sqrt(a)) + x*sqrt(a)*sqrt(b)*sqrt((a - b*x^2)/a))/(2*sqrt(b)), True))
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