Integral de 1/(x^2+9)^3/2 dx

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

   $\int \frac{1}{2 \left(x^{2} + 9\right)^{3}}\, dx = \frac{\int \frac{1}{\left(x^{2} + 9\right)^{3}}\, dx}{2}$

   TrigSubstitutionRule(theta=_theta, func=3*tan(_theta), rewritten=cos(_theta)**4/243, substep=ConstantTimesRule(constant=1/243, other=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), context=cos(_theta)**4/243, symbol=_theta), restriction=True, context=1/((x**2 + 9)**3), symbol=x)

   Por lo tanto, el resultado es: $\frac{x \left(9 - x^{2}\right)}{1296 \left(x^{2} + 9\right)^{2}} + \frac{x}{324 \left(x^{2} + 9\right)} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{1296}$

2. Ahora simplificar:

   $\frac{x^{3}}{432 \left(x^{2} + 9\right)^{2}} + \frac{5 x}{144 \left(x^{2} + 9\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{1296}$

3. Añadimos la constante de integración:

   $\frac{x^{3}}{432 \left(x^{2} + 9\right)^{2}} + \frac{5 x}{144 \left(x^{2} + 9\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{1296}+ \mathrm{constant}$

Respuesta:

$\frac{x^{3}}{432 \left(x^{2} + 9\right)^{2}} + \frac{5 x}{144 \left(x^{2} + 9\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{1296}+ \mathrm{constant}$