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

Optimal. Leaf size=51 \[ -\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{x}{2}-\frac{1}{2} \log (1-x) \]

[Out]

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

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Rubi [A]  time = 0.263505, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28 \[ -\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{x}{2}-\frac{1}{2} \log (1-x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\sqrt{x} \sqrt{- x + 2}}{2} - \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x}}{2} \right )} - \int ^{\sqrt{x}} x\, dx - 2 \int ^{\sqrt{x}} \frac{- \frac{x}{4} - \frac{\sqrt{- x^{2} + 2}}{4}}{x + 1}\, dx - 2 \int ^{\sqrt{x}} \frac{\frac{x}{4} + \frac{\sqrt{- x^{2} + 2}}{4}}{x - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*sqrt(-x + 2)/2 - asin(sqrt(2)*sqrt(x)/2) - Integral(x, (x, sqrt(x))) -
2*Integral((-x/4 - sqrt(-x**2 + 2)/4)/(x + 1), (x, sqrt(x))) - 2*Integral((x/4 +
sqrt(-x**2 + 2)/4)/(x - 1), (x, sqrt(x)))

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Mathematica [A]  time = 0.01792, size = 41, normalized size = 0.8 \[ \frac{1}{2} \left (-x-\sqrt{-(x-2) x}-2 \log (1-x)+\log \left (\sqrt{-(x-2) x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 51, normalized size = 1. \[ -{\frac{1}{2}\sqrt{2-x}\sqrt{x} \left ( \sqrt{-x \left ( x-2 \right ) }-{\it Artanh} \left ({\frac{1}{\sqrt{-x \left ( x-2 \right ) }}} \right ) \right ){\frac{1}{\sqrt{-x \left ( x-2 \right ) }}}}-{\frac{x}{2}}-{\frac{\ln \left ( -1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+(2-x)^(1/2)*x^(1/2))/(2-2*x),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 [A]  time = 0.804424, size = 73, normalized size = 1.43 \[ -\frac{1}{2} \, x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac{2}{{\left | x - 1 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(x + sqrt(x)*sqrt(-x + 2))/(x - 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.304433, size = 86, normalized size = 1.69 \[ -\frac{1}{2} \, x - \frac{1}{2} \, \sqrt{x} \sqrt{-x + 2} - \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (\frac{x + \sqrt{x} \sqrt{-x + 2}}{x}\right ) - \frac{1}{2} \, \log \left (-\frac{x - \sqrt{x} \sqrt{-x + 2}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(x + sqrt(x)*sqrt(-x + 2))/(x - 1),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. \[ - \frac{\int \frac{x}{x - 1}\, dx + \int \frac{\sqrt{x} \sqrt{- x + 2}}{x - 1}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(Integral(x/(x - 1), x) + Integral(sqrt(x)*sqrt(-x + 2)/(x - 1), x))/2

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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(-1/2*(x + sqrt(x)*sqrt(-x + 2))/(x - 1),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError