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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(E^x)/(2*sqrt(x))+sqrt(x)*exp(x)/(1+exp(2*x))

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

Expresión a simplificar:

Solución

Ha introducido [src] 
    / x\     ___  x
atan\E /   \/ x *e 
-------- + --------
    ___         2*x
2*\/ x     1 + e
$$\frac{\sqrt{x} e^{x}}{e^{2 x} + 1} + \frac{\operatorname{atan}{\left(e^{x} \right)}}{2 \sqrt{x}}$$ 
atan(E^x)/((2*sqrt(x))) + (sqrt(x)*exp(x))/(1 + exp(2*x))
Simplificación general [src] 
       /     2*x\     / x\
   x   \1 + e   /*atan\e /
x*e  + -------------------
                2         
--------------------------
       ___ /     2*x\     
     \/ x *\1 + e   /
$$\frac{x e^{x} + \frac{\left(e^{2 x} + 1\right) \operatorname{atan}{\left(e^{x} \right)}}{2}}{\sqrt{x} \left(e^{2 x} + 1\right)}$$ 
(x*exp(x) + (1 + exp(2*x))*atan(exp(x))/2)/(sqrt(x)*(1 + exp(2*x)))
Respuesta numérica [src] 
0.5*x^(-0.5)*atan(E^x) + x^0.5*exp(x)/(1.0 + exp(2*x))
0.5*x^(-0.5)*atan(E^x) + x^0.5*exp(x)/(1.0 + exp(2*x))
Denominador racional [src] 
    / x\  2*x        x       / x\
atan\e /*e    + 2*x*e  + atan\e /
---------------------------------
            ___ /     2*x\       
        2*\/ x *\1 + e   /
$$\frac{2 x e^{x} + e^{2 x} \operatorname{atan}{\left(e^{x} \right)} + \operatorname{atan}{\left(e^{x} \right)}}{2 \sqrt{x} \left(e^{2 x} + 1\right)}$$ 
(atan(exp(x))*exp(2*x) + 2*x*exp(x) + atan(exp(x)))/(2*sqrt(x)*(1 + exp(2*x)))
Combinatoria [src] 
    / x\  2*x        x       / x\
atan\e /*e    + 2*x*e  + atan\e /
---------------------------------
            ___ /     2*x\       
        2*\/ x *\1 + e   /
$$\frac{2 x e^{x} + e^{2 x} \operatorname{atan}{\left(e^{x} \right)} + \operatorname{atan}{\left(e^{x} \right)}}{2 \sqrt{x} \left(e^{2 x} + 1\right)}$$ 
(atan(exp(x))*exp(2*x) + 2*x*exp(x) + atan(exp(x)))/(2*sqrt(x)*(1 + exp(2*x)))
Denominador común [src] 
    / x\  2*x        x       / x\
atan\e /*e    + 2*x*e  + atan\e /
---------------------------------
          ___       ___  2*x     
      2*\/ x  + 2*\/ x *e
$$\frac{2 x e^{x} + e^{2 x} \operatorname{atan}{\left(e^{x} \right)} + \operatorname{atan}{\left(e^{x} \right)}}{2 \sqrt{x} e^{2 x} + 2 \sqrt{x}}$$ 
(atan(exp(x))*exp(2*x) + 2*x*exp(x) + atan(exp(x)))/(2*sqrt(x) + 2*sqrt(x)*exp(2*x))
Parte trigonométrica [src] 
                            ___                    
atan(cosh(x) + sinh(x))   \/ x *(cosh(x) + sinh(x))
----------------------- + -------------------------
            ___           1 + cosh(2*x) + sinh(2*x)
        2*\/ x
$$\frac{\sqrt{x} \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{\sinh{\left(2 x \right)} + \cosh{\left(2 x \right)} + 1} + \frac{\operatorname{atan}{\left(\sinh{\left(x \right)} + \cosh{\left(x \right)} \right)}}{2 \sqrt{x}}$$ 
    /                   x\     ___                    
atan\(cosh(1) + sinh(1)) /   \/ x *(cosh(x) + sinh(x))
-------------------------- + -------------------------
             ___             1 + cosh(2*x) + sinh(2*x)
         2*\/ x
$$\frac{\sqrt{x} \left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right)}{\sinh{\left(2 x \right)} + \cosh{\left(2 x \right)} + 1} + \frac{\operatorname{atan}{\left(\left(\sinh{\left(1 \right)} + \cosh{\left(1 \right)}\right)^{x} \right)}}{2 \sqrt{x}}$$ 
atan((cosh(1) + sinh(1))^x)/(2*sqrt(x)) + sqrt(x)*(cosh(x) + sinh(x))/(1 + cosh(2*x) + sinh(2*x))
Unión de expresiones racionales [src] 
/     2*x\     / x\        x
\1 + e   /*atan\e / + 2*x*e 
----------------------------
         ___ /     2*x\     
     2*\/ x *\1 + e   /
$$\frac{2 x e^{x} + \left(e^{2 x} + 1\right) \operatorname{atan}{\left(e^{x} \right)}}{2 \sqrt{x} \left(e^{2 x} + 1\right)}$$ 
((1 + exp(2*x))*atan(exp(x)) + 2*x*exp(x))/(2*sqrt(x)*(1 + exp(2*x)))
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