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Simplificación de expresiones/ Descomposición en fracciones simples/ exp(x)*sin(pi*x)/(1+pi^2)-pi*cos(pi*x)*exp(x)/(1+pi^2)

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

Expresión a simplificar:

Solución

Ha introducido [src] 
 x                           x
e *sin(pi*x)   pi*cos(pi*x)*e 
------------ - ---------------
        2                2    
  1 + pi           1 + pi
−πcos⁡(πx)ex1+π2+exsin⁡(πx)1+π2- \frac{\pi \cos{\left(\pi x \right)} e^{x}}{1 + \pi^{2}} + \frac{e^{x} \sin{\left(\pi x \right)}}{1 + \pi^{2}} 
(exp(x)*sin(pi*x))/(1 + pi^2) - (pi*cos(pi*x))*exp(x)/(1 + pi^2)
Simplificación general [src] 
                             x
(-pi*cos(pi*x) + sin(pi*x))*e 
------------------------------
                 2            
           1 + pi
(sin⁡(πx)−πcos⁡(πx))ex1+π2\frac{\left(\sin{\left(\pi x \right)} - \pi \cos{\left(\pi x \right)}\right) e^{x}}{1 + \pi^{2}} 
(-pi*cos(pi*x) + sin(pi*x))*exp(x)/(1 + pi^2)
Respuesta numérica [src] 
0.0919996683503752*exp(x)*sin(pi*x) - 0.289025482222236*cos(pi*x)*exp(x)
0.0919996683503752*exp(x)*sin(pi*x) - 0.289025482222236*cos(pi*x)*exp(x)
Potencias [src] 
     / pi*I*x    -pi*I*x\                                 
     |e         e       |  x                              
  pi*|------- + --------|*e      /   -pi*I*x    pi*I*x\  x
     \   2         2    /      I*\- e        + e      /*e 
- -------------------------- - ---------------------------
                 2                       /      2\        
           1 + pi                      2*\1 + pi /
−π(eiπx2+e−iπx2)ex1+π2−i(eiπx−e−iπx)ex2(1+π2)- \frac{\pi \left(\frac{e^{i \pi x}}{2} + \frac{e^{- i \pi x}}{2}\right) e^{x}}{1 + \pi^{2}} - \frac{i \left(e^{i \pi x} - e^{- i \pi x}\right) e^{x}}{2 \left(1 + \pi^{2}\right)} 
 x                           x
e *sin(pi*x)   pi*cos(pi*x)*e 
------------ - ---------------
        2                2    
  1 + pi           1 + pi
exsin⁡(πx)1+π2−πexcos⁡(πx)1+π2\frac{e^{x} \sin{\left(\pi x \right)}}{1 + \pi^{2}} - \frac{\pi e^{x} \cos{\left(\pi x \right)}}{1 + \pi^{2}} 
exp(x)*sin(pi*x)/(1 + pi^2) - pi*cos(pi*x)*exp(x)/(1 + pi^2)
Denominador común [src] 
 /   x                           x\ 
-\- e *sin(pi*x) + pi*cos(pi*x)*e / 
------------------------------------
                    2               
              1 + pi
−−exsin⁡(πx)+πexcos⁡(πx)1+π2- \frac{- e^{x} \sin{\left(\pi x \right)} + \pi e^{x} \cos{\left(\pi x \right)}}{1 + \pi^{2}} 
-(-exp(x)*sin(pi*x) + pi*cos(pi*x)*exp(x))/(1 + pi^2)
Combinatoria [src] 
                              x 
-(-sin(pi*x) + pi*cos(pi*x))*e  
--------------------------------
                  2             
            1 + pi
−(−sin⁡(πx)+πcos⁡(πx))ex1+π2- \frac{\left(- \sin{\left(\pi x \right)} + \pi \cos{\left(\pi x \right)}\right) e^{x}}{1 + \pi^{2}} 
-(-sin(pi*x) + pi*cos(pi*x))*exp(x)/(1 + pi^2)
Denominador racional [src] 
 x                           x
e *sin(pi*x) - pi*cos(pi*x)*e 
------------------------------
                 2            
           1 + pi
exsin⁡(πx)−πexcos⁡(πx)1+π2\frac{e^{x} \sin{\left(\pi x \right)} - \pi e^{x} \cos{\left(\pi x \right)}}{1 + \pi^{2}} 
(exp(x)*sin(pi*x) - pi*cos(pi*x)*exp(x))/(1 + pi^2)
Compilar la expresión [src] 
 x                           x
e *sin(pi*x)   pi*cos(pi*x)*e 
------------ - ---------------
        2                2    
  1 + pi           1 + pi
exsin⁡(πx)1+π2−πexcos⁡(πx)1+π2\frac{e^{x} \sin{\left(\pi x \right)}}{1 + \pi^{2}} - \frac{\pi e^{x} \cos{\left(\pi x \right)}}{1 + \pi^{2}} 
exp(x)*sin(pi*x)/(1 + pi^2) - pi*cos(pi*x)*exp(x)/(1 + pi^2)
Unión de expresiones racionales [src] 
                             x
(-pi*cos(pi*x) + sin(pi*x))*e 
------------------------------
                 2            
           1 + pi
(sin⁡(πx)−πcos⁡(πx))ex1+π2\frac{\left(\sin{\left(\pi x \right)} - \pi \cos{\left(\pi x \right)}\right) e^{x}}{1 + \pi^{2}} 
(-pi*cos(pi*x) + sin(pi*x))*exp(x)/(1 + pi^2)
Parte trigonométrica [src] 
    cosh(x)            sinh(x)        pi*cosh(x)   pi*sinh(x)
---------------- + ---------------- - ---------- - ----------
   /  pi       \      /  pi       \   sec(pi*x)    sec(pi*x) 
sec|- -- + pi*x|   sec|- -- + pi*x|                          
   \  2        /      \  2        /                          
-------------------------------------------------------------
                                 2                           
                           1 + pi
sinh⁡(x)sec⁡(πx−π2)−πsinh⁡(x)sec⁡(πx)+cosh⁡(x)sec⁡(πx−π2)−πcosh⁡(x)sec⁡(πx)1+π2\frac{\frac{\sinh{\left(x \right)}}{\sec{\left(\pi x - \frac{\pi}{2} \right)}} - \frac{\pi \sinh{\left(x \right)}}{\sec{\left(\pi x \right)}} + \frac{\cosh{\left(x \right)}}{\sec{\left(\pi x - \frac{\pi}{2} \right)}} - \frac{\pi \cosh{\left(x \right)}}{\sec{\left(\pi x \right)}}}{1 + \pi^{2}} 
                       /  pi       \                                   
(cosh(x) + sinh(x))*cos|- -- + pi*x|                                   
                       \  2        /   pi*(cosh(x) + sinh(x))*cos(pi*x)
------------------------------------ - --------------------------------
                    2                                    2             
              1 + pi                               1 + pi
−π(sinh⁡(x)+cosh⁡(x))cos⁡(πx)1+π2+(sinh⁡(x)+cosh⁡(x))cos⁡(πx−π2)1+π2- \frac{\pi \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \cos{\left(\pi x \right)}}{1 + \pi^{2}} + \frac{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \cos{\left(\pi x - \frac{\pi}{2} \right)}}{1 + \pi^{2}} 
   /  pi       \              /  pi       \                                                      
cos|- -- + pi*x|*cosh(x) + cos|- -- + pi*x|*sinh(x) - pi*cos(pi*x)*cosh(x) - pi*cos(pi*x)*sinh(x)
   \  2        /              \  2        /                                                      
-------------------------------------------------------------------------------------------------
                                                   2                                             
                                             1 + pi
−πcos⁡(πx)sinh⁡(x)−πcos⁡(πx)cosh⁡(x)+cos⁡(πx−π2)sinh⁡(x)+cos⁡(πx−π2)cosh⁡(x)1+π2\frac{- \pi \cos{\left(\pi x \right)} \sinh{\left(x \right)} - \pi \cos{\left(\pi x \right)} \cosh{\left(x \right)} + \cos{\left(\pi x - \frac{\pi}{2} \right)} \sinh{\left(x \right)} + \cos{\left(\pi x - \frac{\pi}{2} \right)} \cosh{\left(x \right)}}{1 + \pi^{2}} 
             /pi*x\                /pi*x\      /       2/pi*x\\              /       2/pi*x\\        
2*cosh(x)*tan|----|   2*sinh(x)*tan|----|   pi*|1 - tan |----||*cosh(x)   pi*|1 - tan |----||*sinh(x)
             \ 2  /                \ 2  /      \        \ 2  //              \        \ 2  //        
------------------- + ------------------- - --------------------------- - ---------------------------
          2/pi*x\               2/pi*x\                   2/pi*x\                       2/pi*x\      
   1 + tan |----|        1 + tan |----|            1 + tan |----|                1 + tan |----|      
           \ 2  /                \ 2  /                    \ 2  /                        \ 2  /      
-----------------------------------------------------------------------------------------------------
                                                     2                                               
                                               1 + pi
−π(1−tan⁡2(πx2))sinh⁡(x)tan⁡2(πx2)+1−π(1−tan⁡2(πx2))cosh⁡(x)tan⁡2(πx2)+1+2tan⁡(πx2)sinh⁡(x)tan⁡2(πx2)+1+2tan⁡(πx2)cosh⁡(x)tan⁡2(πx2)+11+π2\frac{- \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right) \sinh{\left(x \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1} - \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right) \cosh{\left(x \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{\pi x}{2} \right)} \sinh{\left(x \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{\pi x}{2} \right)} \cosh{\left(x \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}}{1 + \pi^{2}} 
                             x
(-pi*cos(pi*x) + sin(pi*x))*e 
------------------------------
                 2            
           1 + pi
(sin⁡(πx)−πcos⁡(πx))ex1+π2\frac{\left(\sin{\left(\pi x \right)} - \pi \cos{\left(\pi x \right)}\right) e^{x}}{1 + \pi^{2}} 
             x                          x       
            e                       pi*e        
-------------------------- - -------------------
/      2\    /  pi       \   /      2\          
\1 + pi /*sec|- -- + pi*x|   \1 + pi /*sec(pi*x)
             \  2        /
ex(1+π2)sec⁡(πx−π2)−πex(1+π2)sec⁡(πx)\frac{e^{x}}{\left(1 + \pi^{2}\right) \sec{\left(\pi x - \frac{\pi}{2} \right)}} - \frac{\pi e^{x}}{\left(1 + \pi^{2}\right) \sec{\left(\pi x \right)}} 
/       1               pi   \  x
|---------------- - ---------|*e 
|   /  pi       \   sec(pi*x)|   
|sec|- -- + pi*x|            |   
\   \  2        /            /   
---------------------------------
                   2             
             1 + pi
(1sec⁡(πx−π2)−πsec⁡(πx))ex1+π2\frac{\left(\frac{1}{\sec{\left(\pi x - \frac{\pi}{2} \right)}} - \frac{\pi}{\sec{\left(\pi x \right)}}\right) e^{x}}{1 + \pi^{2}} 
          x                      x       
         e                   pi*e        
------------------- - -------------------
/      2\             /      2\          
\1 + pi /*csc(pi*x)   \1 + pi /*sec(pi*x)
−πex(1+π2)sec⁡(πx)+ex(1+π2)csc⁡(πx)- \frac{\pi e^{x}}{\left(1 + \pi^{2}\right) \sec{\left(\pi x \right)}} + \frac{e^{x}}{\left(1 + \pi^{2}\right) \csc{\left(\pi x \right)}} 
          x                        x          
         e                     pi*e           
------------------- - ------------------------
/      2\             /      2\    /pi       \
\1 + pi /*csc(pi*x)   \1 + pi /*csc|-- - pi*x|
                                   \2        /
−πex(1+π2)csc⁡(−πx+π2)+ex(1+π2)csc⁡(πx)- \frac{\pi e^{x}}{\left(1 + \pi^{2}\right) \csc{\left(- \pi x + \frac{\pi}{2} \right)}} + \frac{e^{x}}{\left(1 + \pi^{2}\right) \csc{\left(\pi x \right)}} 
                                                          /pi       \
                                pi*(cosh(x) + sinh(x))*sin|-- + pi*x|
(cosh(x) + sinh(x))*sin(pi*x)                             \2        /
----------------------------- - -------------------------------------
                 2                                   2               
           1 + pi                              1 + pi
(sinh⁡(x)+cosh⁡(x))sin⁡(πx)1+π2−π(sinh⁡(x)+cosh⁡(x))sin⁡(πx+π2)1+π2\frac{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \sin{\left(\pi x \right)}}{1 + \pi^{2}} - \frac{\pi \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \sin{\left(\pi x + \frac{\pi}{2} \right)}}{1 + \pi^{2}} 
 cosh(x)     sinh(x)      pi*cosh(x)       pi*sinh(x)  
--------- + --------- - -------------- - --------------
csc(pi*x)   csc(pi*x)      /pi       \      /pi       \
                        csc|-- - pi*x|   csc|-- - pi*x|
                           \2        /      \2        /
-------------------------------------------------------
                              2                        
                        1 + pi
−πsinh⁡(x)csc⁡(−πx+π2)+sinh⁡(x)csc⁡(πx)−πcosh⁡(x)csc⁡(−πx+π2)+cosh⁡(x)csc⁡(πx)1+π2\frac{- \frac{\pi \sinh{\left(x \right)}}{\csc{\left(- \pi x + \frac{\pi}{2} \right)}} + \frac{\sinh{\left(x \right)}}{\csc{\left(\pi x \right)}} - \frac{\pi \cosh{\left(x \right)}}{\csc{\left(- \pi x + \frac{\pi}{2} \right)}} + \frac{\cosh{\left(x \right)}}{\csc{\left(\pi x \right)}}}{1 + \pi^{2}} 
         x    /pi*x\              /       2/pi*x\\  x  
      2*e *tan|----|           pi*|1 - tan |----||*e   
              \ 2  /              \        \ 2  //     
-------------------------- - --------------------------
/      2\ /       2/pi*x\\   /      2\ /       2/pi*x\\
\1 + pi /*|1 + tan |----||   \1 + pi /*|1 + tan |----||
          \        \ 2  //             \        \ 2  //
−π(1−tan⁡2(πx2))ex(1+π2)(tan⁡2(πx2)+1)+2extan⁡(πx2)(1+π2)(tan⁡2(πx2)+1)- \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right) e^{x}}{\left(1 + \pi^{2}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)} + \frac{2 e^{x} \tan{\left(\frac{\pi x}{2} \right)}}{\left(1 + \pi^{2}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)} 
(cosh(x) + sinh(x))*sin(pi*x)   pi*(cosh(x) + sinh(x))*cos(pi*x)
----------------------------- - --------------------------------
                 2                                2             
           1 + pi                           1 + pi
(sinh⁡(x)+cosh⁡(x))sin⁡(πx)1+π2−π(sinh⁡(x)+cosh⁡(x))cos⁡(πx)1+π2\frac{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \sin{\left(\pi x \right)}}{1 + \pi^{2}} - \frac{\pi \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) \cos{\left(\pi x \right)}}{1 + \pi^{2}} 
/                   /  pi       \\                    
|-pi*cos(pi*x) + cos|- -- + pi*x||*(cosh(x) + sinh(x))
\                   \  2        //                    
------------------------------------------------------
                             2                        
                       1 + pi
(−πcos⁡(πx)+cos⁡(πx−π2))(sinh⁡(x)+cosh⁡(x))1+π2\frac{\left(- \pi \cos{\left(\pi x \right)} + \cos{\left(\pi x - \frac{\pi}{2} \right)}\right) \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{1 + \pi^{2}} 
/      /pi*x\        /        2/pi*x\\\   
| 2*cot|----|     pi*|-1 + cot |----|||   
|      \ 2  /        \         \ 2  //|  x
|-------------- - --------------------|*e 
|       2/pi*x\             2/pi*x\   |   
|1 + cot |----|      1 + cot |----|   |   
\        \ 2  /              \ 2  /   /   
------------------------------------------
                       2                  
                 1 + pi
(−π(cot⁡2(πx2)−1)cot⁡2(πx2)+1+2cot⁡(πx2)cot⁡2(πx2)+1)ex1+π2\frac{\left(- \frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{\pi x}{2} \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1}\right) e^{x}}{1 + \pi^{2}} 
/    1             pi      \  x
|--------- - --------------|*e 
|csc(pi*x)      /pi       \|   
|            csc|-- - pi*x||   
\               \2        //   
-------------------------------
                  2            
            1 + pi
(−πcsc⁡(−πx+π2)+1csc⁡(πx))ex1+π2\frac{\left(- \frac{\pi}{\csc{\left(- \pi x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(\pi x \right)}}\right) e^{x}}{1 + \pi^{2}} 
             /pi*x\        /pi*x\              /        2/pi*x\\              /        2/pi*x\\        
2*cosh(x)*cot|----|   2*cot|----|*sinh(x)   pi*|-1 + cot |----||*cosh(x)   pi*|-1 + cot |----||*sinh(x)
             \ 2  /        \ 2  /              \         \ 2  //              \         \ 2  //        
------------------- + ------------------- - ---------------------------- - ----------------------------
          2/pi*x\               2/pi*x\                   2/pi*x\                        2/pi*x\       
   1 + cot |----|        1 + cot |----|            1 + cot |----|                 1 + cot |----|       
           \ 2  /                \ 2  /                    \ 2  /                         \ 2  /       
-------------------------------------------------------------------------------------------------------
                                                      2                                                
                                                1 + pi
−π(cot⁡2(πx2)−1)sinh⁡(x)cot⁡2(πx2)+1−π(cot⁡2(πx2)−1)cosh⁡(x)cot⁡2(πx2)+1+2cot⁡(πx2)sinh⁡(x)cot⁡2(πx2)+1+2cot⁡(πx2)cosh⁡(x)cot⁡2(πx2)+11+π2\frac{- \frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right) \sinh{\left(x \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1} - \frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right) \cosh{\left(x \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{\pi x}{2} \right)} \sinh{\left(x \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{\pi x}{2} \right)} \cosh{\left(x \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1}}{1 + \pi^{2}} 
(-pi*cos(pi*x) + sin(pi*x))*(cosh(x) + sinh(x))
-----------------------------------------------
                          2                    
                    1 + pi
(sin⁡(πx)−πcos⁡(πx))(sinh⁡(x)+cosh⁡(x))1+π2\frac{\left(\sin{\left(\pi x \right)} - \pi \cos{\left(\pi x \right)}\right) \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{1 + \pi^{2}} 
                                                      /pi       \         /pi       \        
cosh(x)*sin(pi*x) + sin(pi*x)*sinh(x) - pi*cosh(x)*sin|-- + pi*x| - pi*sin|-- + pi*x|*sinh(x)
                                                      \2        /         \2        /        
---------------------------------------------------------------------------------------------
                                                 2                                           
                                           1 + pi
sin⁡(πx)sinh⁡(x)+sin⁡(πx)cosh⁡(x)−πsin⁡(πx+π2)sinh⁡(x)−πsin⁡(πx+π2)cosh⁡(x)1+π2\frac{\sin{\left(\pi x \right)} \sinh{\left(x \right)} + \sin{\left(\pi x \right)} \cosh{\left(x \right)} - \pi \sin{\left(\pi x + \frac{\pi}{2} \right)} \sinh{\left(x \right)} - \pi \sin{\left(\pi x + \frac{\pi}{2} \right)} \cosh{\left(x \right)}}{1 + \pi^{2}} 
/      /pi*x\        /       2/pi*x\\\   
| 2*tan|----|     pi*|1 - tan |----|||   
|      \ 2  /        \        \ 2  //|  x
|-------------- - -------------------|*e 
|       2/pi*x\             2/pi*x\  |   
|1 + tan |----|      1 + tan |----|  |   
\        \ 2  /              \ 2  /  /   
-----------------------------------------
                       2                 
                 1 + pi
(−π(1−tan⁡2(πx2))tan⁡2(πx2)+1+2tan⁡(πx2)tan⁡2(πx2)+1)ex1+π2\frac{\left(- \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right)}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{\pi x}{2} \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}\right) e^{x}}{1 + \pi^{2}} 
cosh(x)*sin(pi*x) + sin(pi*x)*sinh(x) - pi*cos(pi*x)*cosh(x) - pi*cos(pi*x)*sinh(x)
-----------------------------------------------------------------------------------
                                            2                                      
                                      1 + pi
sin⁡(πx)sinh⁡(x)+sin⁡(πx)cosh⁡(x)−πcos⁡(πx)sinh⁡(x)−πcos⁡(πx)cosh⁡(x)1+π2\frac{\sin{\left(\pi x \right)} \sinh{\left(x \right)} + \sin{\left(\pi x \right)} \cosh{\left(x \right)} - \pi \cos{\left(\pi x \right)} \sinh{\left(x \right)} - \pi \cos{\left(\pi x \right)} \cosh{\left(x \right)}}{1 + \pi^{2}} 
/        /pi       \            \                    
|- pi*sin|-- + pi*x| + sin(pi*x)|*(cosh(x) + sinh(x))
\        \2        /            /                    
-----------------------------------------------------
                             2                       
                       1 + pi
(sin⁡(πx)−πsin⁡(πx+π2))(sinh⁡(x)+cosh⁡(x))1+π2\frac{\left(\sin{\left(\pi x \right)} - \pi \sin{\left(\pi x + \frac{\pi}{2} \right)}\right) \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{1 + \pi^{2}} 
           /pi*x\  x             /        2/pi*x\\  x  
      2*cot|----|*e           pi*|-1 + cot |----||*e   
           \ 2  /                \         \ 2  //     
-------------------------- - --------------------------
/      2\ /       2/pi*x\\   /      2\ /       2/pi*x\\
\1 + pi /*|1 + cot |----||   \1 + pi /*|1 + cot |----||
          \        \ 2  //             \        \ 2  //
−π(cot⁡2(πx2)−1)ex(1+π2)(cot⁡2(πx2)+1)+2excot⁡(πx2)(1+π2)(cot⁡2(πx2)+1)- \frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right) e^{x}}{\left(1 + \pi^{2}\right) \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)} + \frac{2 e^{x} \cot{\left(\frac{\pi x}{2} \right)}}{\left(1 + \pi^{2}\right) \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)} 
2*cot(pi*x/2)*exp(x)/((1 + pi^2)*(1 + cot(pi*x/2)^2)) - pi*(-1 + cot(pi*x/2)^2)*exp(x)/((1 + pi^2)*(1 + cot(pi*x/2)^2))
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