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

Optimal. Leaf size=234 \[ \frac{1}{3} x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )+\frac{2 \left (1-x^3\right )^{2/3} x^2}{3 \left (x^3+1\right )}+\frac{\left (1-x^3\right )^{2/3} x}{3 \left (x^3+1\right )}-\frac{\left (1-x^3\right )^{2/3}}{3 \left (x^3+1\right )}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{3 \sqrt [3]{2}}+\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{3 \sqrt [3]{2}}-\frac{2^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3}}-\frac{2^{2/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [F]  time = 0.414851, 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 = {} \[ \int \frac{\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-4*Defer[Int][(1 - x^3)^(2/3)/(1 + I*Sqrt[3] - 2*x)^2, x])/3 + (((4*I)/3)*Defer[Int][(1 - x^3)^(2/3)/(1 + I*S
qrt[3] - 2*x), x])/Sqrt[3] - (4*Defer[Int][(1 - x^3)^(2/3)/(-1 + I*Sqrt[3] + 2*x)^2, x])/3 + (((4*I)/3)*Defer[
Int][(1 - x^3)^(2/3)/(-1 + I*Sqrt[3] + 2*x), x])/Sqrt[3]

Rubi steps

\begin{align*} \int \frac{\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx &=\int \left (-\frac{4 \left (1-x^3\right )^{2/3}}{3 \left (1+i \sqrt{3}-2 x\right )^2}+\frac{4 i \left (1-x^3\right )^{2/3}}{3 \sqrt{3} \left (1+i \sqrt{3}-2 x\right )}-\frac{4 \left (1-x^3\right )^{2/3}}{3 \left (-1+i \sqrt{3}+2 x\right )^2}+\frac{4 i \left (1-x^3\right )^{2/3}}{3 \sqrt{3} \left (-1+i \sqrt{3}+2 x\right )}\right ) \, dx\\ &=-\left (\frac{4}{3} \int \frac{\left (1-x^3\right )^{2/3}}{\left (1+i \sqrt{3}-2 x\right )^2} \, dx\right )-\frac{4}{3} \int \frac{\left (1-x^3\right )^{2/3}}{\left (-1+i \sqrt{3}+2 x\right )^2} \, dx+\frac{(4 i) \int \frac{\left (1-x^3\right )^{2/3}}{1+i \sqrt{3}-2 x} \, dx}{3 \sqrt{3}}+\frac{(4 i) \int \frac{\left (1-x^3\right )^{2/3}}{-1+i \sqrt{3}+2 x} \, dx}{3 \sqrt{3}}\\ \end{align*}

Mathematica [F]  time = 0.45237, size = 0, normalized size = 0. \[ \int \frac{\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({x}^{2}-x+1 \right ) ^{2}} \left ( -{x}^{3}+1 \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x^3 + 1)^(2/3)/(x^2 - x + 1)^2, 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^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-x^3 + 1)^(2/3)/(x^4 - 2*x^3 + 3*x^2 - 2*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}}}{\left (x^{2} - x + 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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