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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    pi       pi*atan(x)
---------- - ----------
  /     2\        2    
x*\1 + x /       x
$$\frac{\pi}{x \left(x^{2} + 1\right)} - \frac{\pi \operatorname{atan}{\left(x \right)}}{x^{2}}$$ 
pi/((x*(1 + x^2))) - pi*atan(x)/x^2
Simplificación general [src] 
   /    /     2\        \
pi*\x - \1 + x /*atan(x)/
-------------------------
        2 /     2\       
       x *\1 + x /
$$\frac{\pi \left(x - \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}\right)}{x^{2} \left(x^{2} + 1\right)}$$ 
pi*(x - (1 + x^2)*atan(x))/(x^2*(1 + x^2))
Respuesta numérica [src] 
3.14159265358979/(x*(1.0 + x^2)) - 3.14159265358979*atan(x)/x^2
3.14159265358979/(x*(1.0 + x^2)) - 3.14159265358979*atan(x)/x^2
Denominador común [src] 
 /                        2        \ 
-\pi*atan(x) - pi*x + pi*x *atan(x)/ 
-------------------------------------
                2    4               
               x  + x
$$- \frac{\pi x^{2} \operatorname{atan}{\left(x \right)} - \pi x + \pi \operatorname{atan}{\left(x \right)}}{x^{4} + x^{2}}$$ 
-(pi*atan(x) - pi*x + pi*x^2*atan(x))/(x^2 + x^4)
Abrimos la expresión [src] 
  pi*atan(x)       pi    
- ---------- + ----------
       2         /     2\
      x        x*\1 + x /
$$- \frac{\pi \operatorname{atan}{\left(x \right)}}{x^{2}} + \frac{\pi}{x \left(x^{2} + 1\right)}$$ 
-pi*atan(x)/x^2 + pi/(x*(1 + x^2))
Denominador racional [src] 
    2        /     2\        
pi*x  - pi*x*\1 + x /*atan(x)
-----------------------------
          3 /     2\         
         x *\1 + x /
$$\frac{\pi x^{2} - \pi x \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}{x^{3} \left(x^{2} + 1\right)}$$ 
(pi*x^2 - pi*x*(1 + x^2)*atan(x))/(x^3*(1 + x^2))
Unión de expresiones racionales [src] 
   /    /     2\        \
pi*\x - \1 + x /*atan(x)/
-------------------------
        2 /     2\       
       x *\1 + x /
$$\frac{\pi \left(x - \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}\right)}{x^{2} \left(x^{2} + 1\right)}$$ 
pi*(x - (1 + x^2)*atan(x))/(x^2*(1 + x^2))
Combinatoria [src] 
    /      2                  \ 
-pi*\-x + x *atan(x) + atan(x)/ 
--------------------------------
           2 /     2\           
          x *\1 + x /
$$- \frac{\pi \left(x^{2} \operatorname{atan}{\left(x \right)} - x + \operatorname{atan}{\left(x \right)}\right)}{x^{2} \left(x^{2} + 1\right)}$$ 
-pi*(-x + x^2*atan(x) + atan(x))/(x^2*(1 + x^2))
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