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

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 = 0.0550118, 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 \[ \frac{1}{3} \sqrt{\frac{1-x^3}{x^3+1}} \left (x^3+1\right )-\frac{2}{3} \tan ^{-1}\left (\sqrt{\frac{1-x^3}{x^3+1}}\right ) \]

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

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Rubi in Sympy [A]  time = 2.41022, size = 46, normalized size = 0.87 \[ \frac{2 \sqrt{\frac{- x^{3} + 1}{x^{3} + 1}}}{3 \left (\frac{- x^{3} + 1}{x^{3} + 1} + 1\right )} - \frac{2 \operatorname{atan}{\left (\sqrt{\frac{- x^{3} + 1}{x^{3} + 1}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*((-x**3+1)/(x**3+1))**(1/2),x)

[Out]

2*sqrt((-x**3 + 1)/(x**3 + 1))/(3*((-x**3 + 1)/(x**3 + 1) + 1)) - 2*atan(sqrt((-
x**3 + 1)/(x**3 + 1)))/3

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Mathematica [A]  time = 0.0491164, size = 79, normalized size = 1.49 \[ \frac{\sqrt{\frac{1-x^3}{x^3+1}} \left (\sqrt{1-x^3} \left (x^3+1\right )+2 \sqrt{x^3+1} \sin ^{-1}\left (\frac{\sqrt{x^3+1}}{\sqrt{2}}\right )\right )}{3 \sqrt{1-x^3}} \]

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

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Maple [A]  time = 0.094, size = 68, normalized size = 1.3 \[{\frac{{x}^{3}+1}{3}\sqrt{-{\frac{{x}^{3}-1}{{x}^{3}+1}}}}-{\frac{\arcsin \left ({x}^{3} \right ) }{3\,{x}^{3}-3}\sqrt{-{\frac{{x}^{3}-1}{{x}^{3}+1}}}\sqrt{- \left ({x}^{3}-1 \right ) \left ({x}^{3}+1 \right ) }} \]

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. \[ \int x^{2} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.267438, size = 117, normalized size = 2.21 \[ -\frac{x^{6} + 2 \,{\left ({\left (x^{3} + 1\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - 1\right )} \arctan \left (\frac{{\left (x^{3} + 1\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - 1}{x^{3}}\right )}{3 \,{\left ({\left (x^{3} + 1\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(x^6 + 2*((x^3 + 1)*sqrt(-(x^3 - 1)/(x^3 + 1)) - 1)*arctan(((x^3 + 1)*sqrt(
-(x^3 - 1)/(x^3 + 1)) - 1)/x^3))/((x^3 + 1)*sqrt(-(x^3 - 1)/(x^3 + 1)) - 1)

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

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/XCAS [A]  time = 0.264491, size = 30, normalized size = 0.57 \[ \frac{1}{3} \,{\left (\sqrt{-x^{6} + 1} + \arcsin \left (x^{3}\right )\right )}{\rm sign}\left (x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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