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

TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=cos(_theta)**6, substep=RewriteRule(rewritten=(cos(2*_theta)/2 + 1/2)**3, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 + 3*cos(2*_theta)/8 + 1/8, substep=AddRule(substeps=[ConstantTimesRule(constant=1/8, other=cos(2*_theta)**3, substep=RewriteRule(rewritten=(1 - sin(2*_theta)**2)*cos(2*_theta), substep=AlternativeRule(alternatives=[AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_u), ConstantTimesRule(constant=-1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2/2, symbol=_u)], context=1/2 - _u**2/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u)/2, symbol=_u), ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/2, symbol=_u)], context=-sin(_u)**2*cos(_u)/2 + cos(_u)/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), context=cos(2*_theta)**3, symbol=_theta), context=cos(2*_theta)**3/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)**2/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)/8, symbol=_theta), ConstantRule(constant=1/8, context=1/8, symbol=_theta)], context=cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 + 3*cos(2*_theta)/8 + 1/8, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**3, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 + 3*cos(2*_theta)/8 + 1/8, substep=AddRule(substeps=[ConstantTimesRule(constant=1/8, other=cos(2*_theta)**3, substep=RewriteRule(rewritten=(1 - sin(2*_theta)**2)*cos(2*_theta), substep=AlternativeRule(alternatives=[AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_u), ConstantTimesRule(constant=-1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2/2, symbol=_u)], context=1/2 - _u**2/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u)/2, symbol=_u), ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/2, symbol=_u)], context=-sin(_u)**2*cos(_u)/2 + cos(_u)/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), context=cos(2*_theta)**3, symbol=_theta), context=cos(2*_theta)**3/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)**2/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)/8, symbol=_theta), ConstantRule(constant=1/8, context=1/8, symbol=_theta)], context=cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 + 3*cos(2*_theta)/8 + 1/8, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**3, symbol=_theta)], context=(cos(2*_theta)/2 + 1/2)**3, symbol=_theta), context=cos(_theta)**6, symbol=_theta), restriction=True, context=1/((x**2 + 1)**4), symbol=x)

1. Ahora simplificar:

   $\frac{- 8 x^{3} - 9 x \left(x^{2} - 1\right) \left(x^{2} + 1\right) + 24 x \left(x^{2} + 1\right)^{2} + 15 \left(x^{2} + 1\right)^{3} \operatorname{atan}{\left(x \right)}}{48 \left(x^{2} + 1\right)^{3}}$

2. Añadimos la constante de integración:

   $\frac{- 8 x^{3} - 9 x \left(x^{2} - 1\right) \left(x^{2} + 1\right) + 24 x \left(x^{2} + 1\right)^{2} + 15 \left(x^{2} + 1\right)^{3} \operatorname{atan}{\left(x \right)}}{48 \left(x^{2} + 1\right)^{3}}+ \mathrm{constant}$

Respuesta:

$\frac{- 8 x^{3} - 9 x \left(x^{2} - 1\right) \left(x^{2} + 1\right) + 24 x \left(x^{2} + 1\right)^{2} + 15 \left(x^{2} + 1\right)^{3} \operatorname{atan}{\left(x \right)}}{48 \left(x^{2} + 1\right)^{3}}+ \mathrm{constant}$