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

Optimal. Leaf size=14 \[ \frac{3}{2} \log \left (1-x^{2/3}\right ) \]

[Out]

(3*Log[1 - x^(2/3)])/2

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Rubi [A]  time = 0.010194, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{3}{2} \log \left (1-x^{2/3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-x^(1/3) + x)^(-1),x]

[Out]

(3*Log[1 - x^(2/3)])/2

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Rubi in Sympy [A]  time = 0.959532, size = 10, normalized size = 0.71 \[ \frac{3 \log{\left (- x^{\frac{2}{3}} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(-x**(2/3) + 1)/2

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Mathematica [A]  time = 0.00567106, size = 14, normalized size = 1. \[ \frac{3}{2} \log \left (1-x^{2/3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*Log[1 - x^(2/3)])/2

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Maple [B]  time = 0.008, size = 50, normalized size = 3.6 \[{\frac{\ln \left ( -1+x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{2}}+\ln \left ( -1+\sqrt [3]{x} \right ) -{\frac{1}{2}\ln \left ({x}^{{\frac{2}{3}}}+\sqrt [3]{x}+1 \right ) }+\ln \left ( \sqrt [3]{x}+1 \right ) -{\frac{1}{2}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^(1/3)+x),x)

[Out]

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

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Maxima [A]  time = 1.36053, size = 23, normalized size = 1.64 \[ \frac{3}{2} \, \log \left (x^{\frac{1}{3}} + 1\right ) + \frac{3}{2} \, \log \left (x^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.198534, size = 11, normalized size = 0.79 \[ \frac{3}{2} \, \log \left (x^{\frac{2}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*log(x^(2/3) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**(1/3)+x),x)

[Out]

Integral(1/(-x**(1/3) + x), x)

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GIAC/XCAS [A]  time = 0.210512, size = 24, normalized size = 1.71 \[ \frac{3}{2} \,{\rm ln}\left (x^{\frac{1}{3}} + 1\right ) + \frac{3}{2} \,{\rm ln}\left ({\left | x^{\frac{1}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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