Integral de x/(x^4+1)^3 dx

1. Hay varias maneras de calcular esta integral.

   Método #1

   1. Vuelva a escribir el integrando:

      $\frac{x}{\left(x^{4} + 1\right)^{3}} = \frac{x}{x^{12} + 3 x^{8} + 3 x^{4} + 1}$

   2. que $u = x^{2}$.

      Luego que $du = 2 x dx$ y ponemos $du$:

      $\int \frac{1}{2 u^{6} + 6 u^{4} + 6 u^{2} + 2}\, du$

      1. Vuelva a escribir el integrando:

         $\frac{1}{2 u^{6} + 6 u^{4} + 6 u^{2} + 2} = \frac{1}{2 \left(u^{2} + 1\right)^{3}}$

      2. La integral del producto de una función por una constante es la constante por la integral de esta función:

         $\int \frac{1}{2 \left(u^{2} + 1\right)^{3}}\, du = \frac{\int \frac{1}{\left(u^{2} + 1\right)^{3}}\, du}{2}$

         TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=cos(_theta)**4, substep=RewriteRule(rewritten=(cos(2*_theta)/2 + 1/2)**2, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta)], context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), context=cos(_theta)**4, symbol=_theta), restriction=True, context=(_u**2 + 1)**(-3), symbol=_u)

         Por lo tanto, el resultado es: $\frac{u \left(1 - u^{2}\right)}{16 \left(u^{2} + 1\right)^{2}} + \frac{u}{4 \left(u^{2} + 1\right)} + \frac{3 \operatorname{atan}{\left(u \right)}}{16}$

      Si ahora sustituir $u$ más en:

      $\frac{x^{2} \left(1 - x^{4}\right)}{16 \left(x^{4} + 1\right)^{2}} + \frac{x^{2}}{4 \left(x^{4} + 1\right)} + \frac{3 \operatorname{atan}{\left(x^{2} \right)}}{16}$

   Método #2

   1. Vuelva a escribir el integrando:

      $\frac{x}{\left(x^{4} + 1\right)^{3}} = \frac{x}{x^{12} + 3 x^{8} + 3 x^{4} + 1}$

   2. que $u = x^{2}$.

      Luego que $du = 2 x dx$ y ponemos $du$:

      $\int \frac{1}{2 u^{6} + 6 u^{4} + 6 u^{2} + 2}\, du$

      1. Vuelva a escribir el integrando:

         $\frac{1}{2 u^{6} + 6 u^{4} + 6 u^{2} + 2} = \frac{1}{2 \left(u^{2} + 1\right)^{3}}$

      2. La integral del producto de una función por una constante es la constante por la integral de esta función:

         $\int \frac{1}{2 \left(u^{2} + 1\right)^{3}}\, du = \frac{\int \frac{1}{\left(u^{2} + 1\right)^{3}}\, du}{2}$

         TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=cos(_theta)**4, substep=RewriteRule(rewritten=(cos(2*_theta)/2 + 1/2)**2, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta)], context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), context=cos(_theta)**4, symbol=_theta), restriction=True, context=(_u**2 + 1)**(-3), symbol=_u)

         Por lo tanto, el resultado es: $\frac{u \left(1 - u^{2}\right)}{16 \left(u^{2} + 1\right)^{2}} + \frac{u}{4 \left(u^{2} + 1\right)} + \frac{3 \operatorname{atan}{\left(u \right)}}{16}$

      Si ahora sustituir $u$ más en:

      $\frac{x^{2} \left(1 - x^{4}\right)}{16 \left(x^{4} + 1\right)^{2}} + \frac{x^{2}}{4 \left(x^{4} + 1\right)} + \frac{3 \operatorname{atan}{\left(x^{2} \right)}}{16}$

2. Ahora simplificar:

   $\frac{3 x^{6}}{16 \left(x^{4} + 1\right)^{2}} + \frac{5 x^{2}}{16 \left(x^{4} + 1\right)^{2}} + \frac{3 \operatorname{atan}{\left(x^{2} \right)}}{16}$

3. Añadimos la constante de integración:

   $\frac{3 x^{6}}{16 \left(x^{4} + 1\right)^{2}} + \frac{5 x^{2}}{16 \left(x^{4} + 1\right)^{2}} + \frac{3 \operatorname{atan}{\left(x^{2} \right)}}{16}+ \mathrm{constant}$

Respuesta:

$\frac{3 x^{6}}{16 \left(x^{4} + 1\right)^{2}} + \frac{5 x^{2}}{16 \left(x^{4} + 1\right)^{2}} + \frac{3 \operatorname{atan}{\left(x^{2} \right)}}{16}+ \mathrm{constant}$