Integral de 1/(x^2+81)^2 dx

TrigSubstitutionRule(theta=_theta, func=9*tan(_theta), rewritten=cos(_theta)**2/729, substep=ConstantTimesRule(constant=1/729, 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=cos(_theta)**2/729, symbol=_theta), restriction=True, context=1/((x**2 + 81)**2), symbol=x)

1. Ahora simplificar:

   $\frac{9 x + \left(x^{2} + 81\right) \operatorname{atan}{\left(\frac{x}{9} \right)}}{1458 \left(x^{2} + 81\right)}$

2. Añadimos la constante de integración:

   $\frac{9 x + \left(x^{2} + 81\right) \operatorname{atan}{\left(\frac{x}{9} \right)}}{1458 \left(x^{2} + 81\right)}+ \mathrm{constant}$

Respuesta:

$\frac{9 x + \left(x^{2} + 81\right) \operatorname{atan}{\left(\frac{x}{9} \right)}}{1458 \left(x^{2} + 81\right)}+ \mathrm{constant}$