3.1017 \(\int \frac{1-x^2}{(1-x+x^2) (1-x^3)^{2/3}} \, dx\)

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) - Log[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 = 0.534065, 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 = {} \[ \int \frac{1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx \]

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*Sqrt[3] + 2*x)*(1 - x^3)^(2/3)), x]

Rubi steps

\begin{align*} \int \frac{1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx &=\int \left (-\frac{1}{\left (1-x^3\right )^{2/3}}+\frac{2-x}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}\right ) \, dx\\ &=-\int \frac{1}{\left (1-x^3\right )^{2/3}} \, dx+\int \frac{2-x}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx\\ &=-x \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};x^3\right )+\int \left (\frac{-1-i \sqrt{3}}{\left (-1-i \sqrt{3}+2 x\right ) \left (1-x^3\right )^{2/3}}+\frac{-1+i \sqrt{3}}{\left (-1+i \sqrt{3}+2 x\right ) \left (1-x^3\right )^{2/3}}\right ) \, dx\\ &=-x \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};x^3\right )+\left (-1-i \sqrt{3}\right ) \int \frac{1}{\left (-1-i \sqrt{3}+2 x\right ) \left (1-x^3\right )^{2/3}} \, dx+\left (-1+i \sqrt{3}\right ) \int \frac{1}{\left (-1+i \sqrt{3}+2 x\right ) \left (1-x^3\right )^{2/3}} \, dx\\ \end{align*}

Mathematica [F]  time = 0.177865, 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 = 0.16, size = 0, normalized size = 0. \begin{align*} \int{\frac{-{x}^{2}+1}{{x}^{2}-x+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+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. \begin{align*} -\int \frac{x^{2} - 1}{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}{\left (x^{2} - x + 1\right )}}\,{d x} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 47.2642, size = 738, normalized size = 7.17 \begin{align*} -\frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{4^{\frac{1}{6}} \sqrt{3}{\left (2 \cdot 4^{\frac{2}{3}}{\left (x^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + x - 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 4 \,{\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} - 4 \, x + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (x^{6} - 7 \, x^{5} + 10 \, x^{4} - 7 \, x^{3} + 10 \, x^{2} - 7 \, x + 1\right )}\right )}}{6 \,{\left (3 \, x^{6} - 9 \, x^{5} + 6 \, x^{4} - x^{3} + 6 \, x^{2} - 9 \, x + 3\right )}}\right ) - \frac{1}{24} \cdot 4^{\frac{2}{3}} \log \left (\frac{2 \cdot 4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}{\left (x^{2} - 3 \, x + 1\right )} - 4^{\frac{2}{3}}{\left (x^{4} - 3 \, x^{2} + 1\right )} - 8 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - x\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac{1}{12} \cdot 4^{\frac{2}{3}} \log \left (-\frac{4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )} - 4^{\frac{1}{3}}{\left (x^{2} - x + 1\right )} - 2 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2} - x + 1}\right ) \end{align*}

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, algorithm="fricas")

[Out]

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(2*4^(2/3)*(x^5 - x^4 - 3*x^3 + 3*x^2 + x - 1)*(-x^3 + 1)^(1/3
) + 4*(x^4 - 4*x^3 + 5*x^2 - 4*x + 1)*(-x^3 + 1)^(2/3) + 4^(1/3)*(x^6 - 7*x^5 + 10*x^4 - 7*x^3 + 10*x^2 - 7*x
+ 1))/(3*x^6 - 9*x^5 + 6*x^4 - x^3 + 6*x^2 - 9*x + 3)) - 1/24*4^(2/3)*log((2*4^(1/3)*(-x^3 + 1)^(2/3)*(x^2 - 3
*x + 1) - 4^(2/3)*(x^4 - 3*x^2 + 1) - 8*(-x^3 + 1)^(1/3)*(x^2 - x))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + 1/12*4^
(2/3)*log(-(4^(2/3)*(-x^3 + 1)^(1/3)*(x - 1) - 4^(1/3)*(x^2 - x + 1) - 2*(-x^3 + 1)^(2/3))/(x^2 - x + 1))

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

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

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

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, algorithm="giac")

[Out]

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