3.4.60 \(\int \frac {1}{-\sqrt {x}+x} \, dx\) [360]

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 = {1607, 266} \begin {gather*} 2 \log \left (1-\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-Sqrt[x] + x)^(-1),x]

[Out]

2*Log[1 - Sqrt[x]]

Rule 266

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 1607

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 = 0.01, size = 10, normalized size = 0.83 \begin {gather*} 2 \log \left (-1+\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-Sqrt[x] + x)^(-1),x]

[Out]

2*Log[-1 + Sqrt[x]]

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Maple [A]
time = 0.36, size = 12, normalized size = 1.00

method	result	size
derivativedivides	\(2 \ln \left (-1+\sqrt {x}\right )\)	\(9\)
meijerg	\(2 \ln \left (1-\sqrt {x}\right )\)	\(11\)
default	\(\ln \left (x -1\right )-2 \arctanh \left (\sqrt {x}\right )\)	\(12\)
trager	\(\ln \left (2 \sqrt {x}-1-x \right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x-x^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 8, normalized size = 0.67 \begin {gather*} 2 \, \log \left (\sqrt {x} - 1\right ) \end {gather*}

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 = 1.40, size = 8, normalized size = 0.67 \begin {gather*} 2 \, \log \left (\sqrt {x} - 1\right ) \end {gather*}

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 = 0.05, size = 8, normalized size = 0.67 \begin {gather*} 2 \log {\left (\sqrt {x} - 1 \right )} \end {gather*}

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 = 0.56, size = 9, normalized size = 0.75 \begin {gather*} 2 \, \log \left ({\left | \sqrt {x} - 1 \right |}\right ) \end {gather*}

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

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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