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

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Rubi [F]  time = 0.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.08, size = 43, normalized size = 1.00 \[ x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {(2 x+1) \left (1-x^3\right )^{2/3}}{x^2+x+1} \]

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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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x - 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}}\,{d x} \]

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)

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maple [A]  time = 0.09, size = 34, normalized size = 0.79 \[ x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )-\frac {\left (x -1\right ) \left (2 x +1\right )}{\left (-x^{3}+1\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x - 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {x-1}{{\left (1-x^3\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{x^{2} \sqrt [3]{1 - x^{3}} + x \sqrt [3]{1 - x^{3}} + \sqrt [3]{1 - x^{3}}}\, dx - \int \left (- \frac {1}{x^{2} \sqrt [3]{1 - x^{3}} + x \sqrt [3]{1 - x^{3}} + \sqrt [3]{1 - x^{3}}}\right )\, dx \]

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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