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

Optimal. Leaf size=20 \[ 3 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right ) \]

[Out]

3*x^(1/3) + 3*Log[1 - x^(1/3)]

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Rubi [A]  time = 0.0202348, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 3 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

3*x^(1/3) + 3*Log[1 - x^(1/3)]

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Rubi in Sympy [A]  time = 1.90405, size = 15, normalized size = 0.75 \[ 3 \sqrt [3]{x} + 3 \log{\left (- \sqrt [3]{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**(1/3) + 3*log(-x**(1/3) + 1)

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Mathematica [A]  time = 0.0093595, size = 20, normalized size = 1. \[ 3 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

3*x^(1/3) + 3*Log[1 - x^(1/3)]

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \[ 3\,\sqrt [3]{x}+3\,\ln \left ( \sqrt [3]{x}-1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^(1/3)+x^(2/3)),x)

[Out]

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

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Maxima [A]  time = 0.716103, size = 19, normalized size = 0.95 \[ 3 \, x^{\frac{1}{3}} + 3 \, \log \left (x^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x^(1/3) + 3*log(x^(1/3) - 1)

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Fricas [A]  time = 0.274374, size = 19, normalized size = 0.95 \[ 3 \, x^{\frac{1}{3}} + 3 \, \log \left (x^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x^(1/3) + 3*log(x^(1/3) - 1)

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Sympy [A]  time = 0.340885, size = 15, normalized size = 0.75 \[ 3 \sqrt [3]{x} + 3 \log{\left (\sqrt [3]{x} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**(1/3) + 3*log(x**(1/3) - 1)

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GIAC/XCAS [A]  time = 0.277788, size = 20, normalized size = 1. \[ 3 \, x^{\frac{1}{3}} + 3 \,{\rm ln}\left ({\left | x^{\frac{1}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x^(1/3) + 3*ln(abs(x^(1/3) - 1))