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 = 0.003826, 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, 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 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]

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]

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*}

Mathematica [A]  time = 0.0022968, 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 = 0.003, size = 12, normalized size = 1. \begin{align*} \ln \left ( -1+x \right ) -2\,{\it Artanh} \left ( \sqrt{x} \right ) \end{align*}

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 = 1.04106, size = 11, normalized size = 0.92 \begin{align*} 2 \, \log \left (\sqrt{x} - 1\right ) \end{align*}

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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Fricas [A]  time = 0.82284, size = 27, normalized size = 2.25 \begin{align*} 2 \, \log \left (\sqrt{x} - 1\right ) \end{align*}

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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Sympy [A]  time = 0.152274, size = 8, normalized size = 0.67 \begin{align*} 2 \log{\left (\sqrt{x} - 1 \right )} \end{align*}

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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Giac [A]  time = 1.17205, size = 12, normalized size = 1. \begin{align*} 2 \, \log \left ({\left | \sqrt{x} - 1 \right |}\right ) \end{align*}

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))