3.59 \(\int \frac{\sqrt [3]{1-x^3}}{1-x+x^2} \, dx\)

Optimal. Leaf size=280 \[ \frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac{3 \log \left (\sqrt [3]{1-x^3}-\sqrt [3]{2} (x-1)\right )}{2\ 2^{2/3}}+\frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{\log \left (\sqrt [3]{1-x^3}+\sqrt [3]{2} x\right )}{2\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} (x-1)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2^{2/3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log \left (-3 (x-1) \left (x^2-x+1\right )\right )}{2\ 2^{2/3}} \]

[Out]

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

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Rubi [F]  time = 0.385211, 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. \[ \text{Int}\left (\frac{\sqrt [3]{1-x^3}}{1-x+x^2},x\right ) \]

Verification is Not applicable to the result.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 \sqrt{3} i \int \frac{\sqrt [3]{- x^{3} + 1}}{2 x - 1 - \sqrt{3} i}\, dx}{3} + \frac{2 \sqrt{3} i \int \frac{\sqrt [3]{- x^{3} + 1}}{2 x - 1 + \sqrt{3} i}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0416973, size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{1-x^3}}{1-x+x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}-x+1}\sqrt [3]{-{x}^{3}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^3+1)^(1/3)/(x^2-x+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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