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

Optimal. Leaf size=24 \[ \sqrt{x-1} \sqrt{x}-\sinh ^{-1}\left (\sqrt{x-1}\right ) \]

[Out]

Sqrt[-1 + x]*Sqrt[x] - ArcSinh[Sqrt[-1 + x]]

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Rubi [A]  time = 0.0349092, antiderivative size = 28, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \sqrt{-\frac{1-x}{x}} x-\tanh ^{-1}\left (\sqrt{-\frac{1-x}{x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(-1 + x)/x],x]

[Out]

Sqrt[-((1 - x)/x)]*x - ArcTanh[Sqrt[-((1 - x)/x)]]

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Rubi in Sympy [A]  time = 1.90904, size = 19, normalized size = 0.79 \[ x \sqrt{1 - \frac{1}{x}} - \operatorname{atanh}{\left (\sqrt{1 - \frac{1}{x}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(1 - 1/x) - atanh(sqrt(1 - 1/x))

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Mathematica [A]  time = 0.0222932, size = 30, normalized size = 1.25 \[ \sqrt{x-1} \sqrt{x}-\log \left (\sqrt{x-1}+\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(-1 + x)/x],x]

[Out]

Sqrt[-1 + x]*Sqrt[x] - Log[Sqrt[-1 + x] + Sqrt[x]]

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Maple [B]  time = 0.008, size = 45, normalized size = 1.9 \[ -{\frac{x}{2}\sqrt{{\frac{-1+x}{x}}} \left ( -2\,\sqrt{{x}^{2}-x}+\ln \left ( x-{\frac{1}{2}}+\sqrt{{x}^{2}-x} \right ) \right ){\frac{1}{\sqrt{x \left ( -1+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((-1+x)/x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.718419, size = 69, normalized size = 2.88 \[ -\frac{\sqrt{\frac{x - 1}{x}}}{\frac{x - 1}{x} - 1} - \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt((x - 1)/x)/((x - 1)/x - 1) - 1/2*log(sqrt((x - 1)/x) + 1) + 1/2*log(sqrt((
x - 1)/x) - 1)

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Fricas [A]  time = 0.268847, size = 54, normalized size = 2.25 \[ x \sqrt{\frac{x - 1}{x}} - \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x - 1}{x}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt((x - 1)/x) - 1/2*log(sqrt((x - 1)/x) + 1) + 1/2*log(sqrt((x - 1)/x) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-1+x)/x)**(1/2),x)

[Out]

Integral(sqrt((x - 1)/x), x)

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GIAC/XCAS [A]  time = 0.267937, size = 47, normalized size = 1.96 \[ \frac{1}{2} \,{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x} + 1 \right |}\right ){\rm sign}\left (x\right ) + \sqrt{x^{2} - x}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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