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

Optimal. Leaf size=43 \[ x^2 \left (-\, _2F_1\left (\frac{2}{3},\frac{4}{3};\frac{5}{3};x^3\right )\right )+\frac{x}{\sqrt [3]{1-x^3}}+\frac{1}{\sqrt [3]{1-x^3}} \]

[Out]

(1 - x^3)^(-1/3) + x/(1 - x^3)^(1/3) - x^2*Hypergeometric2F1[2/3, 4/3, 5/3, x^3]

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Rubi [F]  time = 0.220072, 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 = {} \[ \int \frac{1-x}{\left (1+x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0745515, size = 43, normalized size = 1. \[ x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )+\frac{(2 x+1) \left (1-x^3\right )^{2/3}}{x^2+x+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.027, size = 34, normalized size = 0.8 \begin{align*} -{ \left ( -1+x \right ) \left ( 1+2\,x \right ){\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}+{x}^{2}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{2}{3}};\,{\frac{5}{3}};\,{x}^{3})} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1+x)*(1+2*x)/(-x^3+1)^(1/3)+x^2*hypergeom([1/3,2/3],[5/3],x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((x - 1)/((-x^3 + 1)^(1/3)*(x^2 + x + 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^{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((1-x)/(x^2+x+1)/(-x^3+1)^(1/3),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{x}{x^{2} \sqrt [3]{1 - x^{3}} + x \sqrt [3]{1 - x^{3}} + \sqrt [3]{1 - x^{3}}}\, dx - \int - \frac{1}{x^{2} \sqrt [3]{1 - x^{3}} + x \sqrt [3]{1 - x^{3}} + \sqrt [3]{1 - x^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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