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

Optimal. Leaf size=177 \[ \frac{1}{2} x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )+\frac{1}{2} \left (1-x^3\right )^{2/3}-\frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{\sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log \left ((1-x) (x+1)^2\right )}{2 \sqrt [3]{2}} \]

[Out]

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

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Rubi [F]  time = 0.0706373, 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 = {} \[ \int \frac{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.150094, size = 0, normalized size = 0. \[ \int \frac{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.011, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}-x+1}{{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((x^2-x+1)*(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int((x^2-x+1)*(-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^{2} - x + 1\right )}}{x^{3} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x^3 + 1)^(2/3)*(x^2 - 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}}}{x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-x^3 + 1)^(2/3)/(x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-(x - 1)*(x**2 + x + 1))**(2/3)/(x + 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^{2} - x + 1\right )}}{x^{3} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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