3.360 \(\int \frac{1}{-\sqrt{x}+x} \, dx\)

Optimal. Leaf size=12 \[ 2 \log \left (1-\sqrt{x}\right ) \]

[Out]

2*Log[1 - Sqrt[x]]

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Rubi [A]  time = 0.0103588, antiderivative size = 12, 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 \[ 2 \log \left (1-\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-Sqrt[x] + x)^(-1),x]

[Out]

2*Log[1 - Sqrt[x]]

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Rubi in Sympy [A]  time = 1.94399, size = 8, normalized size = 0.67 \[ 2 \log{\left (- \sqrt{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(-sqrt(x) + 1)

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Mathematica [A]  time = 0.00661213, size = 12, normalized size = 1. \[ 2 \log \left (1-\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Log[1 - Sqrt[x]]

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Maple [A]  time = 0.006, size = 12, normalized size = 1. \[ \ln \left ( -1+x \right ) -2\,{\it Artanh} \left ( \sqrt{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x-x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.37828, size = 11, normalized size = 0.92 \[ 2 \, \log \left (\sqrt{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(sqrt(x) - 1)

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Fricas [A]  time = 0.222104, size = 11, normalized size = 0.92 \[ 2 \, \log \left (\sqrt{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(sqrt(x) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x-x**(1/2)),x)

[Out]

Integral(1/(-sqrt(x) + x), x)

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GIAC/XCAS [A]  time = 0.220904, size = 12, normalized size = 1. \[ 2 \,{\rm ln}\left ({\left | \sqrt{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*ln(abs(sqrt(x) - 1))