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

Optimal. Leaf size=383 \[ \frac{1}{2} x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )-\frac{\log \left (x^3+1\right )}{3 \sqrt [3]{2}}-\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac{1}{3} 2^{2/3} \log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )+\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{\sqrt [3]{2}}-\frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac{\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}-\frac{2^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{2^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log \left ((1-x) (x+1)^2\right )}{6 \sqrt [3]{2}} \]

[Out]

-((2^(2/3)*ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3]) - ArcTan[(1 + (2^(1/3)*(1 - x))
/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3]) + ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/Sqrt[3] - (2^(2/3)*
ArcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3] + (x^2*Hypergeometric2F1[1/3, 2/3, 5/3, x^3])/2 -
Log[(1 - x)*(1 + x)^2]/(6*2^(1/3)) - Log[1 + x^3]/(3*2^(1/3)) - Log[1 + (2^(2/3)*(1 - x)^2)/(1 - x^3)^(2/3) -
(2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(3*2^(1/3)) + (2^(2/3)*Log[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)])/3 + Log
[-(2^(1/3)*x) - (1 - x^3)^(1/3)]/2^(1/3) - Log[x + (1 - x^3)^(1/3)]/2 + Log[-1 + x + 2^(2/3)*(1 - x^3)^(1/3)]/
(2*2^(1/3))

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Rubi [F]  time = 0.398948, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*Defer[Int][(1 - x^3)^(2/3)/(-1 - x), x])/3 - ((1 + (-1)^(2/3))*Defer[Int][(1 - x^3)^(2/3)/(-1 + (-1)^(1/3)
*x), x])/3 - ((1 - (-1)^(1/3))*Defer[Int][(1 - x^3)^(2/3)/(-1 - (-1)^(2/3)*x), x])/3

Rubi steps

\begin{align*} \int \frac{(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx &=\int \left (-\frac{2 \left (1-x^3\right )^{2/3}}{3 (-1-x)}+\frac{\left (-1-(-1)^{2/3}\right ) \left (1-x^3\right )^{2/3}}{3 \left (-1+\sqrt [3]{-1} x\right )}+\frac{\left (-1+\sqrt [3]{-1}\right ) \left (1-x^3\right )^{2/3}}{3 \left (-1-(-1)^{2/3} x\right )}\right ) \, dx\\ &=-\left (\frac{2}{3} \int \frac{\left (1-x^3\right )^{2/3}}{-1-x} \, dx\right )+\frac{1}{3} \left (-1+\sqrt [3]{-1}\right ) \int \frac{\left (1-x^3\right )^{2/3}}{-1-(-1)^{2/3} x} \, dx+\frac{1}{3} \left (-1-(-1)^{2/3}\right ) \int \frac{\left (1-x^3\right )^{2/3}}{-1+\sqrt [3]{-1} x} \, dx\\ \end{align*}

Mathematica [C]  time = 0.15278, size = 138, normalized size = 0.36 \[ -\frac{1}{2} x^2 F_1\left (\frac{2}{3};-\frac{2}{3},1;\frac{5}{3};x^3,-x^3\right )-\frac{4 \left (1-x^3\right )^{2/3} x F_1\left (\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{4}{3};-\frac{2}{3},2;\frac{7}{3};x^3,-x^3\right )+2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(x^2*AppellF1[2/3, -2/3, 1, 5/3, x^3, -x^3])/2 - (4*x*(1 - x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3, -x^3])
/((1 + x^3)*(-4*AppellF1[1/3, -2/3, 1, 4/3, x^3, -x^3] + x^3*(3*AppellF1[4/3, -2/3, 2, 7/3, x^3, -x^3] + 2*App
ellF1[4/3, 1/3, 1, 7/3, x^3, -x^3])))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{1-x}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}}{x^{3} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}{\left (x - 1\right )}}{x^{3} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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