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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(a*atan(x))/((1+e^(pi*x))*(1+x^2)^(a/2))

¿Cómo vas a descomponer esta sin(a*atan(x))/((1+e^(pi*x))*(1+x^2)^(a/2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    sin(a*atan(x))   
---------------------
                    a
                    -
                    2
/     pi*x\ /     2\ 
\1 + E    /*\1 + x /
sin⁡(aatan⁡(x))(eπx+1)(x2+1)a2\frac{\sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{\left(e^{\pi x} + 1\right) \left(x^{2} + 1\right)^{\frac{a}{2}}} 
sin(a*atan(x))/(((1 + E^(pi*x))*(1 + x^2)^(a/2)))
Simplificación general [src] 
        -a                
        ---               
         2                
/     2\                  
\1 + x /   *sin(a*atan(x))
--------------------------
             pi*x         
        1 + e
(x2+1)−a2sin⁡(aatan⁡(x))eπx+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{e^{\pi x} + 1} 
(1 + x^2)^(-a/2)*sin(a*atan(x))/(1 + exp(pi*x))
Respuesta numérica [src] 
(1.0 + x^2)^(-0.5*a)*sin(a*atan(x))/(1.0 + 2.71828182845905^(3.14159265358979*x))
(1.0 + x^2)^(-0.5*a)*sin(a*atan(x))/(1.0 + 2.71828182845905^(3.14159265358979*x))
Denominador común [src] 
       sin(a*atan(x))      
---------------------------
        a           a      
        -           -      
        2           2      
/     2\    /     2\   pi*x
\1 + x /  + \1 + x / *e
sin⁡(aatan⁡(x))(x2+1)a2eπx+(x2+1)a2\frac{\sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{\left(x^{2} + 1\right)^{\frac{a}{2}} e^{\pi x} + \left(x^{2} + 1\right)^{\frac{a}{2}}} 
sin(a*atan(x))/((1 + x^2)^(a/2) + (1 + x^2)^(a/2)*exp(pi*x))
Combinatoria [src] 
        -a                
        ---               
         2                
/     2\                  
\1 + x /   *sin(a*atan(x))
--------------------------
             pi*x         
        1 + e
(x2+1)−a2sin⁡(aatan⁡(x))eπx+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{e^{\pi x} + 1} 
(1 + x^2)^(-a/2)*sin(a*atan(x))/(1 + exp(pi*x))
Unión de expresiones racionales [src] 
        -a                
        ---               
         2                
/     2\                  
\1 + x /   *sin(a*atan(x))
--------------------------
             pi*x         
        1 + e
(x2+1)−a2sin⁡(aatan⁡(x))eπx+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{e^{\pi x} + 1} 
(1 + x^2)^(-a/2)*sin(a*atan(x))/(1 + exp(pi*x))
Potencias [src] 
           -a                                   
           ---                                  
            2                                   
   /     2\    /   -I*a*atan(x)    I*a*atan(x)\ 
-I*\1 + x /   *\- e             + e           / 
------------------------------------------------
                   /     pi*x\                  
                 2*\1 + e    /
−i(x2+1)−a2(eiaatan⁡(x)−e−iaatan⁡(x))2(eπx+1)- \frac{i \left(x^{2} + 1\right)^{- \frac{a}{2}} \left(e^{i a \operatorname{atan}{\left(x \right)}} - e^{- i a \operatorname{atan}{\left(x \right)}}\right)}{2 \left(e^{\pi x} + 1\right)} 
        -a                
        ---               
         2                
/     2\                  
\1 + x /   *sin(a*atan(x))
--------------------------
             pi*x         
        1 + e
(x2+1)−a2sin⁡(aatan⁡(x))eπx+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{e^{\pi x} + 1} 
(1 + x^2)^(-a/2)*sin(a*atan(x))/(1 + exp(pi*x))
Abrimos la expresión [src] 
        -a                
        ---               
         2                
/     2\                  
\1 + x /   *sin(a*atan(x))
--------------------------
             pi*x         
        1 + E
(x2+1)−a2sin⁡(aatan⁡(x))eπx+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{e^{\pi x} + 1} 
       sin(a*atan(x))      
---------------------------
        a           a      
        -           -      
        2           2      
/     2\    /     2\   pi*x
\1 + x /  + \1 + x / *e
sin⁡(aatan⁡(x))(x2+1)a2eπx+(x2+1)a2\frac{\sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{\left(x^{2} + 1\right)^{\frac{a}{2}} e^{\pi x} + \left(x^{2} + 1\right)^{\frac{a}{2}}} 
sin(a*atan(x))/((1 + x^2)^(a/2) + (1 + x^2)^(a/2)*exp(pi*x))
Compilar la expresión [src] 
        -a                
        ---               
         2                
/     2\                  
\1 + x /   *sin(a*atan(x))
--------------------------
             pi*x         
        1 + e
(x2+1)−a2sin⁡(aatan⁡(x))eπx+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{e^{\pi x} + 1} 
(1 + x^2)^(-a/2)*sin(a*atan(x))/(1 + exp(pi*x))
Denominador racional [src] 
    ___________                          
   /         a          -a               
  /  /     2\   /     2\                 
\/   \1 + x /  *\1 + x /  *sin(a*atan(x))
-----------------------------------------
                     pi*x                
                1 + e
(x2+1)−a(x2+1)asin⁡(aatan⁡(x))eπx+1\frac{\left(x^{2} + 1\right)^{- a} \sqrt{\left(x^{2} + 1\right)^{a}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{e^{\pi x} + 1} 
sqrt((1 + x^2)^a)*(1 + x^2)^(-a)*sin(a*atan(x))/(1 + exp(pi*x))
Parte trigonométrica [src] 
             -a                  
             ---                 
              2                  
     /     2\       /a*atan(x)\  
   2*\1 + x /   *tan|---------|  
                    \    2    /  
---------------------------------
/       2/a*atan(x)\\ /     pi*x\
|1 + tan |---------||*\1 + e    /
\        \    2    //
2(x2+1)−a2tan⁡(aatan⁡(x)2)(eπx+1)(tan⁡2(aatan⁡(x)2)+1)\frac{2 \left(x^{2} + 1\right)^{- \frac{a}{2}} \tan{\left(\frac{a \operatorname{atan}{\left(x \right)}}{2} \right)}}{\left(e^{\pi x} + 1\right) \left(\tan^{2}{\left(\frac{a \operatorname{atan}{\left(x \right)}}{2} \right)} + 1\right)} 
         -a                
         ---               
          2                
 /     2\                  
 \1 + x /   *sin(a*atan(x))
---------------------------
1 + cosh(pi*x) + sinh(pi*x)
(x2+1)−a2sin⁡(aatan⁡(x))sinh⁡(πx)+cosh⁡(πx)+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{\sinh{\left(\pi x \right)} + \cosh{\left(\pi x \right)} + 1} 
               -a         
               ---        
                2         
       /     2\           
       \1 + x /           
--------------------------
/     pi*x\               
\1 + e    /*csc(a*atan(x))
(x2+1)−a2(eπx+1)csc⁡(aatan⁡(x))\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}}}{\left(e^{\pi x} + 1\right) \csc{\left(a \operatorname{atan}{\left(x \right)} \right)}} 
        -a                       
        ---                      
         2                       
/     2\       /  pi            \
\1 + x /   *cos|- -- + a*atan(x)|
               \  2             /
---------------------------------
                          pi*x   
   1 + (cosh(1) + sinh(1))
(x2+1)−a2cos⁡(aatan⁡(x)−π2)(sinh⁡(1)+cosh⁡(1))πx+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \cos{\left(a \operatorname{atan}{\left(x \right)} - \frac{\pi}{2} \right)}}{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\pi x} + 1} 
                      -a                           
                      ---                          
                       2                           
              /     2\       /a*atan(x)\           
            2*\1 + x /   *tan|---------|           
                             \    2    /           
---------------------------------------------------
/                       pi*x\ /       2/a*atan(x)\\
\1 + (cosh(1) + sinh(1))    /*|1 + tan |---------||
                              \        \    2    //
2(x2+1)−a2tan⁡(aatan⁡(x)2)((sinh⁡(1)+cosh⁡(1))πx+1)(tan⁡2(aatan⁡(x)2)+1)\frac{2 \left(x^{2} + 1\right)^{- \frac{a}{2}} \tan{\left(\frac{a \operatorname{atan}{\left(x \right)}}{2} \right)}}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\pi x} + 1\right) \left(\tan^{2}{\left(\frac{a \operatorname{atan}{\left(x \right)}}{2} \right)} + 1\right)} 
             -a                  
             ---                 
              2                  
     /     2\       /a*atan(x)\  
   2*\1 + x /   *cot|---------|  
                    \    2    /  
---------------------------------
/       2/a*atan(x)\\ /     pi*x\
|1 + cot |---------||*\1 + e    /
\        \    2    //
2(x2+1)−a2cot⁡(aatan⁡(x)2)(eπx+1)(cot⁡2(aatan⁡(x)2)+1)\frac{2 \left(x^{2} + 1\right)^{- \frac{a}{2}} \cot{\left(\frac{a \operatorname{atan}{\left(x \right)}}{2} \right)}}{\left(e^{\pi x} + 1\right) \left(\cot^{2}{\left(\frac{a \operatorname{atan}{\left(x \right)}}{2} \right)} + 1\right)} 
         -a                
         ---               
          2                
 /     2\                  
 \1 + x /   *sin(a*atan(x))
---------------------------
                       pi*x
1 + (cosh(1) + sinh(1))
(x2+1)−a2sin⁡(aatan⁡(x))(sinh⁡(1)+cosh⁡(1))πx+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\pi x} + 1} 
        -a                
        ---               
         2                
/     2\                  
\1 + x /                  
-----------*sin(a*atan(x))
      pi*x                
 1 + E
(x2+1)−a2eπx+1sin⁡(aatan⁡(x))\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}}}{e^{\pi x} + 1} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)} 
                   -a            
                   ---           
                    2            
           /     2\              
           \1 + x /              
---------------------------------
/     pi*x\    /  pi            \
\1 + e    /*sec|- -- + a*atan(x)|
               \  2             /
(x2+1)−a2(eπx+1)sec⁡(aatan⁡(x)−π2)\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}}}{\left(e^{\pi x} + 1\right) \sec{\left(a \operatorname{atan}{\left(x \right)} - \frac{\pi}{2} \right)}} 
                            -a                     
                            ---                    
                             2                     
                    /     2\                       
                    \1 + x /                       
---------------------------------------------------
/                       pi*x\    /  pi            \
\1 + (cosh(1) + sinh(1))    /*sec|- -- + a*atan(x)|
                                 \  2             /
(x2+1)−a2((sinh⁡(1)+cosh⁡(1))πx+1)sec⁡(aatan⁡(x)−π2)\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}}}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\pi x} + 1\right) \sec{\left(a \operatorname{atan}{\left(x \right)} - \frac{\pi}{2} \right)}} 
        -a                
        ---               
         2                
/     2\                  
\1 + x /   *sin(a*atan(x))
--------------------------
             pi*x         
        1 + e
(x2+1)−a2sin⁡(aatan⁡(x))eπx+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \sin{\left(a \operatorname{atan}{\left(x \right)} \right)}}{e^{\pi x} + 1} 
                      -a                           
                      ---                          
                       2                           
              /     2\       /a*atan(x)\           
            2*\1 + x /   *cot|---------|           
                             \    2    /           
---------------------------------------------------
/                       pi*x\ /       2/a*atan(x)\\
\1 + (cosh(1) + sinh(1))    /*|1 + cot |---------||
                              \        \    2    //
2(x2+1)−a2cot⁡(aatan⁡(x)2)((sinh⁡(1)+cosh⁡(1))πx+1)(cot⁡2(aatan⁡(x)2)+1)\frac{2 \left(x^{2} + 1\right)^{- \frac{a}{2}} \cot{\left(\frac{a \operatorname{atan}{\left(x \right)}}{2} \right)}}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\pi x} + 1\right) \left(\cot^{2}{\left(\frac{a \operatorname{atan}{\left(x \right)}}{2} \right)} + 1\right)} 
        -a                       
        ---                      
         2                       
/     2\       /  pi            \
\1 + x /   *cos|- -- + a*atan(x)|
               \  2             /
---------------------------------
   1 + cosh(pi*x) + sinh(pi*x)
(x2+1)−a2cos⁡(aatan⁡(x)−π2)sinh⁡(πx)+cosh⁡(πx)+1\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}} \cos{\left(a \operatorname{atan}{\left(x \right)} - \frac{\pi}{2} \right)}}{\sinh{\left(\pi x \right)} + \cosh{\left(\pi x \right)} + 1} 
                        -a                  
                        ---                 
                         2                  
                /     2\                    
                \1 + x /                    
--------------------------------------------
/                       pi*x\               
\1 + (cosh(1) + sinh(1))    /*csc(a*atan(x))
(x2+1)−a2((sinh⁡(1)+cosh⁡(1))πx+1)csc⁡(aatan⁡(x))\frac{\left(x^{2} + 1\right)^{- \frac{a}{2}}}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{\pi x} + 1\right) \csc{\left(a \operatorname{atan}{\left(x \right)} \right)}} 
(1 + x^2)^(-a/2)/((1 + (cosh(1) + sinh(1))^(pi*x))*csc(a*atan(x)))
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