3.574 \(\int \frac{x^6}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=226 \[ -\frac{1}{3} \left (1-x^3\right )^{2/3} x-\frac{2}{9} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )+\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac{2 \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{1}{9} \log \left (-\frac{x}{\sqrt [3]{1-x^3}}+\frac{x^2}{\left (1-x^3\right )^{2/3}}+1\right )-\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.384678, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{1}{3} \left (1-x^3\right )^{2/3} x-\frac{2}{9} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )+\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac{2 \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{1}{9} \log \left (-\frac{x}{\sqrt [3]{1-x^3}}+\frac{x^2}{\left (1-x^3\right )^{2/3}}+1\right )-\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/((1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 36.2785, size = 206, normalized size = 0.91 \[ - \frac{x}{3 \sqrt [3]{- x^{3} + 1} \left (\frac{x^{3}}{- x^{3} + 1} + 1\right )} - \frac{2 \log{\left (\frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{9} + \frac{2^{\frac{2}{3}} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} + \frac{\log{\left (\frac{x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{9} - \frac{2^{\frac{2}{3}} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{- x^{3} + 1}} - \frac{1}{3}\right ) \right )}}{9} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

-x/(3*(-x**3 + 1)**(1/3)*(x**3/(-x**3 + 1) + 1)) - 2*log(x/(-x**3 + 1)**(1/3) +
1)/9 + 2**(2/3)*log(2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/6 + log(x**2/(-x**3 + 1)*
*(2/3) - x/(-x**3 + 1)**(1/3) + 1)/9 - 2**(2/3)*log(2**(2/3)*x**2/(-x**3 + 1)**(
2/3) - 2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/12 - 2*sqrt(3)*atan(sqrt(3)*(2*x/(3*(-
x**3 + 1)**(1/3)) - 1/3))/9 - 2**(2/3)*sqrt(3)*atan(sqrt(3)*(-2*2**(1/3)*x/(3*(-
x**3 + 1)**(1/3)) + 1/3))/6

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Mathematica [C]  time = 0.453987, size = 233, normalized size = 1.03 \[ \frac{1}{36} \left (\frac{42 x^4 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right )}{\sqrt [3]{1-x^3} \left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};x^3,-x^3\right )-F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};x^3,-x^3\right )\right )-7 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right )\right )}-12 \left (1-x^3\right )^{2/3} x+2^{2/3} \left (2 \log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt{3}}\right )-\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac{2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^6/((1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

(-12*x*(1 - x^3)^(2/3) + (42*x^4*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3])/((1 - x^
3)^(1/3)*(1 + x^3)*(-7*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3] + x^3*(3*AppellF1[7
/3, 1/3, 2, 10/3, x^3, -x^3] - AppellF1[7/3, 4/3, 1, 10/3, x^3, -x^3]))) + 2^(2/
3)*(2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] - Log[1 + (2
^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] + 2*Log[1 + (2^(1/3
)*x)/(-1 + x^3)^(1/3)]))/36

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int{\frac{{x}^{6}}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-x^3+1)^(1/3)/(x^3+1),x)

[Out]

int(x^6/(-x^3+1)^(1/3)/(x^3+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="maxima")

[Out]

integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(1/3)), x)

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Fricas [A]  time = 0.219793, size = 301, normalized size = 1.33 \[ -\frac{1}{108} \, \sqrt{3} 2^{\frac{2}{3}}{\left (6 \, \sqrt{3} 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x + 4 \, \sqrt{3} 2^{\frac{1}{3}} \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) - 2 \, \sqrt{3} 2^{\frac{1}{3}} \log \left (\frac{x^{2} -{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 6 \, \sqrt{3} \log \left (\frac{2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2 \, x}{x}\right ) + 3 \, \sqrt{3} \log \left (-\frac{2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x - 2 \, x^{2} - 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 12 \cdot 2^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right ) + 18 \, \arctan \left (\frac{\sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{3} x}{3 \, x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="fricas")

[Out]

-1/108*sqrt(3)*2^(2/3)*(6*sqrt(3)*2^(1/3)*(-x^3 + 1)^(2/3)*x + 4*sqrt(3)*2^(1/3)
*log((x + (-x^3 + 1)^(1/3))/x) - 2*sqrt(3)*2^(1/3)*log((x^2 - (-x^3 + 1)^(1/3)*x
+ (-x^3 + 1)^(2/3))/x^2) - 6*sqrt(3)*log((2^(2/3)*(-x^3 + 1)^(1/3) + 2*x)/x) +
3*sqrt(3)*log(-(2^(2/3)*(-x^3 + 1)^(1/3)*x - 2*x^2 - 2^(1/3)*(-x^3 + 1)^(2/3))/x
^2) - 12*2^(1/3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) + 18*ar
ctan(1/3*(sqrt(3)*2^(2/3)*(-x^3 + 1)^(1/3) - sqrt(3)*x)/x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

Integral(x**6/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="giac")

[Out]

integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(1/3)), x)