3.529 \(\int x^8 \sqrt{\frac{1-x^3}{1+x^3}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{1}{9} \left (\frac{1-x^3}{x^3+1}\right )^{3/2} \left (x^3+1\right )^3-\frac{1}{6} \sqrt{\frac{1-x^3}{x^3+1}} \left (x^3+1\right )^2+\frac{1}{2} \sqrt{\frac{1-x^3}{x^3+1}} \left (x^3+1\right )-\frac{1}{3} \tan ^{-1}\left (\sqrt{\frac{1-x^3}{x^3+1}}\right ) \]

[Out]

(Sqrt[(1 - x^3)/(1 + x^3)]*(1 + x^3))/2 - (Sqrt[(1 - x^3)/(1 + x^3)]*(1 + x^3)^2
)/6 - (((1 - x^3)/(1 + x^3))^(3/2)*(1 + x^3)^3)/9 - ArcTan[Sqrt[(1 - x^3)/(1 + x
^3)]]/3

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Rubi [A]  time = 0.151415, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{1}{9} \left (\frac{1-x^3}{x^3+1}\right )^{3/2} \left (x^3+1\right )^3-\frac{1}{6} \sqrt{\frac{1-x^3}{x^3+1}} \left (x^3+1\right )^2+\frac{1}{2} \sqrt{\frac{1-x^3}{x^3+1}} \left (x^3+1\right )-\frac{1}{3} \tan ^{-1}\left (\sqrt{\frac{1-x^3}{x^3+1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^8*Sqrt[(1 - x^3)/(1 + x^3)],x]

[Out]

(Sqrt[(1 - x^3)/(1 + x^3)]*(1 + x^3))/2 - (Sqrt[(1 - x^3)/(1 + x^3)]*(1 + x^3)^2
)/6 - (((1 - x^3)/(1 + x^3))^(3/2)*(1 + x^3)^3)/9 - ArcTan[Sqrt[(1 - x^3)/(1 + x
^3)]]/3

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Rubi in Sympy [A]  time = 9.90826, size = 102, normalized size = 0.9 \[ - \frac{8 \left (\frac{- x^{3} + 1}{x^{3} + 1}\right )^{\frac{3}{2}}}{9 \left (\frac{- x^{3} + 1}{x^{3} + 1} + 1\right )^{3}} + \frac{\sqrt{\frac{- x^{3} + 1}{x^{3} + 1}}}{\frac{- x^{3} + 1}{x^{3} + 1} + 1} - \frac{2 \sqrt{\frac{- x^{3} + 1}{x^{3} + 1}}}{3 \left (\frac{- x^{3} + 1}{x^{3} + 1} + 1\right )^{2}} - \frac{\operatorname{atan}{\left (\sqrt{\frac{- x^{3} + 1}{x^{3} + 1}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*((-x**3+1)/(x**3+1))**(1/2),x)

[Out]

-8*((-x**3 + 1)/(x**3 + 1))**(3/2)/(9*((-x**3 + 1)/(x**3 + 1) + 1)**3) + sqrt((-
x**3 + 1)/(x**3 + 1))/((-x**3 + 1)/(x**3 + 1) + 1) - 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 = 0.0844269, size = 86, normalized size = 0.76 \[ \frac{\sqrt{\frac{1-x^3}{x^3+1}} \sqrt{x^3+1} \left (6 \sin ^{-1}\left (\frac{\sqrt{x^3+1}}{\sqrt{2}}\right )+\sqrt{1-x^6} \left (2 x^6-3 x^3+4\right )\right )}{18 \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*Sqrt[(1 - x^3)/(1 + x^3)],x]

[Out]

(Sqrt[(1 - x^3)/(1 + x^3)]*Sqrt[1 + x^3]*(Sqrt[1 - x^6]*(4 - 3*x^3 + 2*x^6) + 6*
ArcSin[Sqrt[1 + x^3]/Sqrt[2]]))/(18*Sqrt[1 - x^3])

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Maple [A]  time = 0.085, size = 80, normalized size = 0.7 \[{\frac{ \left ( 2\,{x}^{6}-3\,{x}^{3}+4 \right ) \left ({x}^{3}+1 \right ) }{18}\sqrt{-{\frac{{x}^{3}-1}{{x}^{3}+1}}}}-{\frac{\arcsin \left ({x}^{3} \right ) }{6\,{x}^{3}-6}\sqrt{-{\frac{{x}^{3}-1}{{x}^{3}+1}}}\sqrt{- \left ({x}^{3}-1 \right ) \left ({x}^{3}+1 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*((-x^3+1)/(x^3+1))^(1/2),x)

[Out]

1/18*(2*x^6-3*x^3+4)*(x^3+1)*(-(x^3-1)/(x^3+1))^(1/2)-1/6*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. \[ \int x^{8} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8*sqrt(-(x^3 - 1)/(x^3 + 1)),x, algorithm="maxima")

[Out]

integrate(x^8*sqrt(-(x^3 - 1)/(x^3 + 1)), x)

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Fricas [A]  time = 0.269401, size = 244, normalized size = 2.16 \[ \frac{2 \, x^{18} - 3 \, x^{15} - 6 \, x^{12} + 15 \, x^{9} - 12 \, x^{3} - 6 \,{\left (3 \, x^{6} -{\left (x^{9} + x^{6} - 4 \, x^{3} - 4\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - 4\right )} \arctan \left (\frac{{\left (x^{3} + 1\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - 1}{x^{3}}\right ) + 3 \,{\left (2 \, x^{15} - x^{12} - 3 \, x^{9} + 4 \, x^{6} + 4 \, x^{3}\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}}}{18 \,{\left (3 \, x^{6} -{\left (x^{9} + x^{6} - 4 \, x^{3} - 4\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8*sqrt(-(x^3 - 1)/(x^3 + 1)),x, algorithm="fricas")

[Out]

1/18*(2*x^18 - 3*x^15 - 6*x^12 + 15*x^9 - 12*x^3 - 6*(3*x^6 - (x^9 + x^6 - 4*x^3
- 4)*sqrt(-(x^3 - 1)/(x^3 + 1)) - 4)*arctan(((x^3 + 1)*sqrt(-(x^3 - 1)/(x^3 + 1
)) - 1)/x^3) + 3*(2*x^15 - x^12 - 3*x^9 + 4*x^6 + 4*x^3)*sqrt(-(x^3 - 1)/(x^3 +
1)))/(3*x^6 - (x^9 + x^6 - 4*x^3 - 4)*sqrt(-(x^3 - 1)/(x^3 + 1)) - 4)

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{8} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8*sqrt(-(x^3 - 1)/(x^3 + 1)),x, algorithm="giac")

[Out]

integrate(x^8*sqrt(-(x^3 - 1)/(x^3 + 1)), x)