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

Optimal. Leaf size=103 \[ -\frac{\log \left (-x^3+2 (1-x)^3+1\right )}{2\ 2^{2/3}}+\frac{3 \log \left (\sqrt [3]{1-x^3}+\sqrt [3]{2} (1-x)\right )}{2\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3}} \]

[Out]

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

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Rubi [F]  time = 0.807854, 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. \[ \text{Int}\left (\frac{1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}},x\right ) \]

Verification is Not applicable to the result.

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

[Out]

-(x*Hypergeometric2F1[1/3, 2/3, 4/3, x^3]) - (1 + I*Sqrt[3])*Defer[Int][1/((-1 -
I*Sqrt[3] + 2*x)*(1 - x^3)^(2/3)), x] - (1 - I*Sqrt[3])*Defer[Int][1/((-1 + I*S
qrt[3] + 2*x)*(1 - x^3)^(2/3)), x]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.119075, size = 0, normalized size = 0. \[ \int \frac{1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.187, size = 0, normalized size = 0. \[ \int{\frac{-{x}^{2}+1}{{x}^{2}-x+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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