3.2575 \(\int \frac{1}{\sqrt [3]{1-x} \sqrt [3]{1+x+x^2}} \, dx\)

Optimal. Leaf size=109 \[ \frac{\sqrt [3]{1-x^3} \log \left (\sqrt [3]{1-x^3}+x\right )}{2 \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}}-\frac{\sqrt [3]{1-x^3} \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}} \]

[Out]

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

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Rubi [A]  time = 0.0586705, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\sqrt [3]{1-x^3} \log \left (\sqrt [3]{1-x^3}+x\right )}{2 \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}}-\frac{\sqrt [3]{1-x^3} \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{1-x} \sqrt [3]{x^2+x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.7023, size = 148, normalized size = 1.36 \[ \frac{\left (- x + 1\right )^{\frac{2}{3}} \left (x^{2} + x + 1\right )^{\frac{2}{3}} \log{\left (\frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{3 \left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\left (- x + 1\right )^{\frac{2}{3}} \left (x^{2} + x + 1\right )^{\frac{2}{3}} \log{\left (\frac{x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6 \left (- x^{3} + 1\right )^{\frac{2}{3}}} + \frac{\sqrt{3} \left (- x + 1\right )^{\frac{2}{3}} \left (x^{2} + x + 1\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{- x^{3} + 1}} - \frac{1}{3}\right ) \right )}}{3 \left (- x^{3} + 1\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-x + 1)**(2/3)*(x**2 + x + 1)**(2/3)*log(x/(-x**3 + 1)**(1/3) + 1)/(3*(-x**3 +
1)**(2/3)) - (-x + 1)**(2/3)*(x**2 + x + 1)**(2/3)*log(x**2/(-x**3 + 1)**(2/3) -
x/(-x**3 + 1)**(1/3) + 1)/(6*(-x**3 + 1)**(2/3)) + sqrt(3)*(-x + 1)**(2/3)*(x**
2 + x + 1)**(2/3)*atan(sqrt(3)*(2*x/(3*(-x**3 + 1)**(1/3)) - 1/3))/(3*(-x**3 + 1
)**(2/3))

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Mathematica [C]  time = 0.208942, size = 280, normalized size = 2.57 \[ \frac{45 (1-x)^{2/3} \left (2 i x+\sqrt{3}+i\right ) \left (2 x+i \sqrt{3}+1\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{2 i (x-1)}{3 i+\sqrt{3}},\frac{2 i (x-1)}{-3 i+\sqrt{3}}\right )}{4 \left (x^2+x+1\right )^{4/3} \left ((x-1) \left (\left (\sqrt{3}+3 i\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};-\frac{2 i (x-1)}{3 i+\sqrt{3}},\frac{2 i (x-1)}{-3 i+\sqrt{3}}\right )-\left (\sqrt{3}-3 i\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};-\frac{2 i (x-1)}{3 i+\sqrt{3}},\frac{2 i (x-1)}{-3 i+\sqrt{3}}\right )\right )-30 i F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{2 i (x-1)}{3 i+\sqrt{3}},\frac{2 i (x-1)}{-3 i+\sqrt{3}}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(45*(1 - x)^(2/3)*(I + Sqrt[3] + (2*I)*x)*(1 + I*Sqrt[3] + 2*x)*AppellF1[2/3, 1/
3, 1/3, 5/3, ((-2*I)*(-1 + x))/(3*I + Sqrt[3]), ((2*I)*(-1 + x))/(-3*I + Sqrt[3]
)])/(4*(1 + x + x^2)^(4/3)*((-30*I)*AppellF1[2/3, 1/3, 1/3, 5/3, ((-2*I)*(-1 + x
))/(3*I + Sqrt[3]), ((2*I)*(-1 + x))/(-3*I + Sqrt[3])] + (-1 + x)*((3*I + Sqrt[3
])*AppellF1[5/3, 1/3, 4/3, 8/3, ((-2*I)*(-1 + x))/(3*I + Sqrt[3]), ((2*I)*(-1 +
x))/(-3*I + Sqrt[3])] - (-3*I + Sqrt[3])*AppellF1[5/3, 4/3, 1/3, 8/3, ((-2*I)*(-
1 + x))/(3*I + Sqrt[3]), ((2*I)*(-1 + x))/(-3*I + Sqrt[3])])))

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Maple [F]  time = 0.189, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [3]{1-x}}}{\frac{1}{\sqrt [3]{{x}^{2}+x+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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