Integral de 1/(x^4+6*x^2+9)dx dx

1. Vuelva a escribir el integrando:

   $\frac{1}{\left(x^{4} + 6 x^{2}\right) + 9} = \frac{1}{\left(x^{2} + 3\right)^{2}}$

   TrigSubstitutionRule(theta=_theta, func=sqrt(3)*tan(_theta), rewritten=sqrt(3)*cos(_theta)**2/9, substep=ConstantTimesRule(constant=sqrt(3)/9, other=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[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=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), context=sqrt(3)*cos(_theta)**2/9, symbol=_theta), restriction=True, context=(x**2 + 3)**(-2), symbol=x)

2. Ahora simplificar:

   $\frac{\sqrt{3} \left(\sqrt{3} x + \left(x^{2} + 3\right) \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}\right)}{18 \left(x^{2} + 3\right)}$

3. Añadimos la constante de integración:

   $\frac{\sqrt{3} \left(\sqrt{3} x + \left(x^{2} + 3\right) \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}\right)}{18 \left(x^{2} + 3\right)}+ \mathrm{constant}$

Respuesta:

$\frac{\sqrt{3} \left(\sqrt{3} x + \left(x^{2} + 3\right) \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}\right)}{18 \left(x^{2} + 3\right)}+ \mathrm{constant}$