Integral de x^3*sqrt(2*x^2-1) dx

TrigSubstitutionRule(theta=_theta, func=sqrt(2)*sec(_theta)/2, rewritten=tan(_theta)**2*sec(_theta)**4/4, substep=ConstantTimesRule(constant=1/4, other=tan(_theta)**2*sec(_theta)**4, substep=RewriteRule(rewritten=(tan(_theta)**2 + 1)*tan(_theta)**2*sec(_theta)**2, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=tan(_theta), constant=1, substep=AddRule(substeps=[PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), PowerRule(base=_u, exp=2, context=_u**2, symbol=_u)], context=_u**4 + _u**2, symbol=_u), context=(tan(_theta)**2 + 1)*tan(_theta)**2*sec(_theta)**2, symbol=_theta), RewriteRule(rewritten=tan(_theta)**4*sec(_theta)**2 + tan(_theta)**2*sec(_theta)**2, substep=AddRule(substeps=[URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)**4*sec(_theta)**2, symbol=_theta), URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)**2*sec(_theta)**2, symbol=_theta)], context=tan(_theta)**4*sec(_theta)**2 + tan(_theta)**2*sec(_theta)**2, symbol=_theta), context=(tan(_theta)**2 + 1)*tan(_theta)**2*sec(_theta)**2, symbol=_theta), RewriteRule(rewritten=tan(_theta)**4*sec(_theta)**2 + tan(_theta)**2*sec(_theta)**2, substep=AddRule(substeps=[URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)**4*sec(_theta)**2, symbol=_theta), URule(u_var=_u, u_func=tan(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)**2*sec(_theta)**2, symbol=_theta)], context=tan(_theta)**4*sec(_theta)**2 + tan(_theta)**2*sec(_theta)**2, symbol=_theta), context=(tan(_theta)**2 + 1)*tan(_theta)**2*sec(_theta)**2, symbol=_theta)], context=(tan(_theta)**2 + 1)*tan(_theta)**2*sec(_theta)**2, symbol=_theta), context=tan(_theta)**2*sec(_theta)**4, symbol=_theta), context=tan(_theta)**2*sec(_theta)**4/4, symbol=_theta), restriction=(x > -sqrt(2)/2) & (x < sqrt(2)/2), context=x**3*sqrt(2*x**2 - 1), symbol=x)

1. Ahora simplificar:

   $\begin{cases} \frac{\left(2 x^{2} - 1\right)^{\frac{3}{2}} \left(3 x^{2} + 1\right)}{30} & \text{for}\: x > - \frac{\sqrt{2}}{2} \wedge x < \frac{\sqrt{2}}{2} \end{cases}$

2. Añadimos la constante de integración:

   $\begin{cases} \frac{\left(2 x^{2} - 1\right)^{\frac{3}{2}} \left(3 x^{2} + 1\right)}{30} & \text{for}\: x > - \frac{\sqrt{2}}{2} \wedge x < \frac{\sqrt{2}}{2} \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} \frac{\left(2 x^{2} - 1\right)^{\frac{3}{2}} \left(3 x^{2} + 1\right)}{30} & \text{for}\: x > - \frac{\sqrt{2}}{2} \wedge x < \frac{\sqrt{2}}{2} \end{cases}+ \mathrm{constant}$