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

Optimal. Leaf size=30 \[ -3 x^{2/3}-x-6 \sqrt [3]{x}-6 \log \left (1-\sqrt [3]{x}\right ) \]

[Out]

-6*x^(1/3) - 3*x^(2/3) - x - 6*Log[1 - x^(1/3)]

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Rubi [A]  time = 0.0549251, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -3 x^{2/3}-x-6 \sqrt [3]{x}-6 \log \left (1-\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-6*x^(1/3) - 3*x^(2/3) - x - 6*Log[1 - x^(1/3)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 6 \sqrt [3]{x} - x - 6 \log{\left (- \sqrt [3]{x} + 1 \right )} - 6 \int ^{\sqrt [3]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*x**(1/3) - x - 6*log(-x**(1/3) + 1) - 6*Integral(x, (x, x**(1/3)))

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Mathematica [A]  time = 0.0105902, size = 30, normalized size = 1. \[ -3 x^{2/3}-x-6 \sqrt [3]{x}-6 \log \left (1-\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-6*x^(1/3) - 3*x^(2/3) - x - 6*Log[1 - x^(1/3)]

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Maple [A]  time = 0.004, size = 23, normalized size = 0.8 \[ -x-3\,{x}^{2/3}-6\,\sqrt [3]{x}-6\,\ln \left ( -1+\sqrt [3]{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x-3*x^(2/3)-6*x^(1/3)-6*ln(-1+x^(1/3))

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Maxima [A]  time = 1.37639, size = 30, normalized size = 1. \[ -x - 3 \, x^{\frac{2}{3}} - 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x - 3*x^(2/3) - 6*x^(1/3) - 6*log(x^(1/3) - 1)

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Fricas [A]  time = 0.2368, size = 30, normalized size = 1. \[ -x - 3 \, x^{\frac{2}{3}} - 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x - 3*x^(2/3) - 6*x^(1/3) - 6*log(x^(1/3) - 1)

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Sympy [A]  time = 0.405284, size = 26, normalized size = 0.87 \[ - 3 x^{\frac{2}{3}} - 6 \sqrt [3]{x} - x - 6 \log{\left (\sqrt [3]{x} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x**(2/3) - 6*x**(1/3) - x - 6*log(x**(1/3) - 1)

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GIAC/XCAS [A]  time = 0.215655, size = 31, normalized size = 1.03 \[ -x - 3 \, x^{\frac{2}{3}} - 6 \, x^{\frac{1}{3}} - 6 \,{\rm ln}\left ({\left | x^{\frac{1}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x - 3*x^(2/3) - 6*x^(1/3) - 6*ln(abs(x^(1/3) - 1))