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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1593, 260} \[ 2 \log \left (1-\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-Sqrt[x] + x)^(-1),x]

[Out]

2*Log[1 - Sqrt[x]]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {1}{-\sqrt {x}+x} \, dx &=\int \frac {1}{\left (-1+\sqrt {x}\right ) \sqrt {x}} \, dx\\ &=2 \log \left (1-\sqrt {x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ 2 \log \left (1-\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Sqrt[x] + x)^(-1),x]

[Out]

2*Log[1 - Sqrt[x]]

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fricas [A]  time = 0.39, size = 8, normalized size = 0.67 \[ 2 \, \log \left (\sqrt {x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(sqrt(x) - 1)

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giac [A]  time = 0.15, size = 9, normalized size = 0.75 \[ 2 \, \log \left ({\left | \sqrt {x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x-x^(1/2)),x, algorithm="giac")

[Out]

2*log(abs(sqrt(x) - 1))

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maple [A]  time = 0.05, size = 12, normalized size = 1.00 \[ -2 \arctanh \left (\sqrt {x}\right )+\ln \left (x -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 8, normalized size = 0.67 \[ 2 \, \log \left (\sqrt {x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x-x^(1/2)),x, algorithm="maxima")

[Out]

2*log(sqrt(x) - 1)

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mupad [B]  time = 0.10, size = 8, normalized size = 0.67 \[ 2\,\ln \left (\sqrt {x}-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - x^(1/2)),x)

[Out]

2*log(x^(1/2) - 1)

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sympy [A]  time = 0.18, size = 8, normalized size = 0.67 \[ 2 \log {\left (\sqrt {x} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x-x**(1/2)),x)

[Out]

2*log(sqrt(x) - 1)

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