3.729 \(\int \frac{\sqrt{x}}{\sqrt{2-x}-\sqrt{x}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{x}{2}-\frac{1}{2} \sqrt{2-x} \sqrt{x}-\frac{1}{2} \log (1-x)+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2-x} \sqrt{x}\right ) \]

[Out]

-(Sqrt[2 - x]*Sqrt[x])/2 - x/2 + ArcTanh[Sqrt[2 - x]*Sqrt[x]]/2 - Log[1 - x]/2

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Rubi [A]  time = 0.11844, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{x}{2}-\frac{1}{2} \sqrt{2-x} \sqrt{x}-\frac{1}{2} \log (1-x)+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2-x} \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(Sqrt[2 - x] - Sqrt[x]),x]

[Out]

-(Sqrt[2 - x]*Sqrt[x])/2 - x/2 + ArcTanh[Sqrt[2 - x]*Sqrt[x]]/2 - Log[1 - x]/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\sqrt{x} \sqrt{- x + 2}}{2} - \frac{\log{\left (- x + 1 \right )}}{2} + \frac{\operatorname{atanh}{\left (\sqrt{x} \sqrt{- x + 2} \right )}}{2} + \int \left (- \frac{1}{2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*sqrt(-x + 2)/2 - log(-x + 1)/2 + atanh(sqrt(x)*sqrt(-x + 2))/2 + Integr
al(-1/2, x)

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Mathematica [A]  time = 0.0488636, size = 86, normalized size = 1.59 \[ \frac{1}{2} \left (-x-\sqrt{-(x-2) x}-\log \left (1-\sqrt{x}\right )+\log \left (\sqrt{2-x}-\sqrt{x}+2\right )+\log \left (\sqrt{x}+1\right )-\log \left (\sqrt{2-x}+\sqrt{x}+2\right )-\log (1-x)\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[x]/(Sqrt[2 - x] - Sqrt[x]),x]

[Out]

(-x - Sqrt[-((-2 + x)*x)] - Log[1 - Sqrt[x]] + Log[2 + Sqrt[2 - x] - Sqrt[x]] +
Log[1 + Sqrt[x]] - Log[2 + Sqrt[2 - x] + Sqrt[x]] - Log[1 - x])/2

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Maple [A]  time = 0.009, size = 51, normalized size = 0.9 \[ -{\frac{1}{2}\sqrt{2-x}\sqrt{x} \left ( \sqrt{-x \left ( x-2 \right ) }-{\it Artanh} \left ({\frac{1}{\sqrt{-x \left ( x-2 \right ) }}} \right ) \right ){\frac{1}{\sqrt{-x \left ( x-2 \right ) }}}}-{\frac{x}{2}}-{\frac{\ln \left ( -1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/((2-x)^(1/2)-x^(1/2)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{x}}{\sqrt{x} - \sqrt{-x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x)/(sqrt(x) - sqrt(-x + 2)),x, algorithm="maxima")

[Out]

-integrate(sqrt(x)/(sqrt(x) - sqrt(-x + 2)), x)

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Fricas [A]  time = 0.276113, size = 86, normalized size = 1.59 \[ -\frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x} \sqrt{-x + 2} - \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (\frac{x + \sqrt{x} \sqrt{-x + 2}}{x}\right ) - \frac{1}{2} \, \log \left (-\frac{x - \sqrt{x} \sqrt{-x + 2}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x)/(sqrt(x) - sqrt(-x + 2)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{- \sqrt{x} + \sqrt{- x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x)/(-sqrt(x) + sqrt(-x + 2)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x)/(sqrt(x) - sqrt(-x + 2)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError