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