3.291 \(\int x^2 \sqrt{\frac{1-x^3}{1+x^3}} \, dx\)

Optimal. Leaf size=53 \[ \frac{1}{3} \sqrt{\frac{1-x^3}{x^3+1}} \left (x^3+1\right )-\frac{2}{3} \tan ^{-1}\left (\sqrt{\frac{1-x^3}{x^3+1}}\right ) \]

[Out]

(Sqrt[(1 - x^3)/(1 + x^3)]*(1 + x^3))/3 - (2*ArcTan[Sqrt[(1 - x^3)/(1 + x^3)]])/3

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Rubi [A]  time = 0.0291266, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1960, 288, 204} \[ \frac{1}{3} \sqrt{\frac{1-x^3}{x^3+1}} \left (x^3+1\right )-\frac{2}{3} \tan ^{-1}\left (\sqrt{\frac{1-x^3}{x^3+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[(1 - x^3)/(1 + x^3)],x]

[Out]

(Sqrt[(1 - x^3)/(1 + x^3)]*(1 + x^3))/3 - (2*ArcTan[Sqrt[(1 - x^3)/(1 + x^3)]])/3

Rule 1960

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(Simplify[(m + 1)/n] - 1
))/(b*e - d*x^q)^(Simplify[(m + 1)/n] + 1), x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, m, n}, x] && FractionQ[p] && IntegerQ[Simplify[(m + 1)/n]]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^2 \sqrt{\frac{1-x^3}{1+x^3}} \, dx &=-\left (\frac{4}{3} \operatorname{Subst}\left (\int \frac{x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt{\frac{1-x^3}{1+x^3}}\right )\right )\\ &=\frac{1}{3} \sqrt{\frac{1-x^3}{1+x^3}} \left (1+x^3\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{\frac{1-x^3}{1+x^3}}\right )\\ &=\frac{1}{3} \sqrt{\frac{1-x^3}{1+x^3}} \left (1+x^3\right )-\frac{2}{3} \tan ^{-1}\left (\sqrt{\frac{1-x^3}{1+x^3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0317817, size = 86, normalized size = 1.62 \[ \frac{\sqrt{\frac{1-x^3}{x^3+1}} \sqrt{x^3+1} \left (\sqrt{x^3+1} \left (x^3-1\right )+2 \sqrt{1-x^3} \sin ^{-1}\left (\frac{\sqrt{1-x^3}}{\sqrt{2}}\right )\right )}{3 \left (x^3-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[(1 - x^3)/(1 + x^3)],x]

[Out]

(Sqrt[(1 - x^3)/(1 + x^3)]*Sqrt[1 + x^3]*((-1 + x^3)*Sqrt[1 + x^3] + 2*Sqrt[1 - x^3]*ArcSin[Sqrt[1 - x^3]/Sqrt
[2]]))/(3*(-1 + x^3))

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Maple [A]  time = 0.081, size = 68, normalized size = 1.3 \begin{align*}{\frac{{x}^{3}+1}{3}\sqrt{-{\frac{{x}^{3}-1}{{x}^{3}+1}}}}-{\frac{\arcsin \left ({x}^{3} \right ) }{3\,{x}^{3}-3}\sqrt{-{\frac{{x}^{3}-1}{{x}^{3}+1}}}\sqrt{- \left ({x}^{3}+1 \right ) \left ({x}^{3}-1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*((-x^3+1)/(x^3+1))^(1/2),x)

[Out]

1/3*(x^3+1)*(-(x^3-1)/(x^3+1))^(1/2)-1/3*arcsin(x^3)*(-(x^3-1)/(x^3+1))^(1/2)*(-(x^3+1)*(x^3-1))^(1/2)/(x^3-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*((-x^3+1)/(x^3+1))^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2*sqrt(-(x^3 - 1)/(x^3 + 1)), x)

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Fricas [A]  time = 1.46181, size = 139, normalized size = 2.62 \begin{align*} \frac{1}{3} \,{\left (x^{3} + 1\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - \frac{2}{3} \, \arctan \left (\frac{{\left (x^{3} + 1\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - 1}{x^{3}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*((-x^3+1)/(x^3+1))^(1/2),x, algorithm="fricas")

[Out]

1/3*(x^3 + 1)*sqrt(-(x^3 - 1)/(x^3 + 1)) - 2/3*arctan(((x^3 + 1)*sqrt(-(x^3 - 1)/(x^3 + 1)) - 1)/x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*((-x**3+1)/(x**3+1))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.17868, size = 30, normalized size = 0.57 \begin{align*} \frac{1}{3} \,{\left (\sqrt{-x^{6} + 1} + \arcsin \left (x^{3}\right )\right )} \mathrm{sgn}\left (x^{3} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*((-x^3+1)/(x^3+1))^(1/2),x, algorithm="giac")

[Out]

1/3*(sqrt(-x^6 + 1) + arcsin(x^3))*sgn(x^3 + 1)