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

Optimal. Leaf size=60 \[ -\frac{x^2}{4}+\frac{1}{4} \sqrt{2-x^2} x-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{2-x^2}}\right )+\frac{1}{4} \log (1-x)+\frac{1}{4} \log (x+1) \]

[Out]

-x^2/4 + (x*Sqrt[2 - x^2])/4 - ArcTanh[x/Sqrt[2 - x^2]]/2 + Log[1 - x]/4 + Log[1
+ x]/4

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Rubi [A]  time = 0.518817, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ -\frac{x^2}{4}+\frac{1}{4} \sqrt{2-x^2} x-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{2-x^2}}\right )+\frac{1}{4} \log (1-x)+\frac{1}{4} \log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

-x^2/4 + (x*Sqrt[2 - x^2])/4 - ArcTanh[x/Sqrt[2 - x^2]]/2 + Log[1 - x]/4 + Log[1
+ x]/4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0296679, size = 77, normalized size = 1.28 \[ \frac{1}{4} \left (-x^2+\sqrt{2-x^2} x+\log \left (1-x^2\right )-\log \left (\sqrt{2-x^2}-x+2\right )+\log \left (\sqrt{2-x^2}+x+2\right )+\log (1-x)-\log (x+1)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-x^2 + x*Sqrt[2 - x^2] + Log[1 - x] - Log[1 + x] + Log[1 - x^2] - Log[2 - x + S
qrt[2 - x^2]] + Log[2 + x + Sqrt[2 - x^2]])/4

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Maple [B]  time = 0.007, size = 111, normalized size = 1.9 \[{\frac{x}{4}\sqrt{-{x}^{2}+2}}-{\frac{1}{4}\sqrt{- \left ( 1+x \right ) ^{2}+3+2\,x}}+{\frac{1}{4}{\it Artanh} \left ({\frac{4+2\,x}{2}{\frac{1}{\sqrt{- \left ( 1+x \right ) ^{2}+3+2\,x}}}} \right ) }+{\frac{1}{4}\sqrt{- \left ( -1+x \right ) ^{2}-2\,x+3}}-{\frac{1}{4}{\it Artanh} \left ({\frac{-2\,x+4}{2}{\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}-2\,x+3}}}} \right ) }-{\frac{{x}^{2}}{4}}+{\frac{\ln \left ( -1+x \right ) }{4}}+{\frac{\ln \left ( 1+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x*(-x^2+2)^(1/2)-1/4*(-(1+x)^2+3+2*x)^(1/2)+1/4*arctanh(1/2*(4+2*x)/(-(1+x)^
2+3+2*x)^(1/2))+1/4*(-(-1+x)^2-2*x+3)^(1/2)-1/4*arctanh(1/2*(-2*x+4)/(-(-1+x)^2-
2*x+3)^(1/2))-1/4*x^2+1/4*ln(-1+x)+1/4*ln(1+x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{2} \, x^{2} - \int -\frac{x^{2}}{x - \sqrt{-x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x^2 - integrate(-x^2/(x - sqrt(-x^2 + 2)), x)

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Fricas [A]  time = 0.273947, size = 90, normalized size = 1.5 \[ -\frac{1}{4} \, x^{2} + \frac{1}{4} \, \sqrt{-x^{2} + 2} x + \frac{1}{4} \, \log \left (x^{2} - 1\right ) - \frac{1}{8} \, \log \left (-\frac{\sqrt{-x^{2} + 2} x + 1}{x^{2}}\right ) + \frac{1}{8} \, \log \left (\frac{\sqrt{-x^{2} + 2} x - 1}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*x^2 + 1/4*sqrt(-x^2 + 2)*x + 1/4*log(x^2 - 1) - 1/8*log(-(sqrt(-x^2 + 2)*x
+ 1)/x^2) + 1/8*log((sqrt(-x^2 + 2)*x - 1)/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.288509, size = 158, normalized size = 2.63 \[ -\frac{1}{4} \, x^{2} + \frac{1}{4} \, \sqrt{-x^{2} + 2} x + \frac{1}{4} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | \frac{x}{\sqrt{2} - \sqrt{-x^{2} + 2}} - \frac{\sqrt{2} - \sqrt{-x^{2} + 2}}{x} + 2 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | \frac{x}{\sqrt{2} - \sqrt{-x^{2} + 2}} - \frac{\sqrt{2} - \sqrt{-x^{2} + 2}}{x} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*x^2 + 1/4*sqrt(-x^2 + 2)*x + 1/4*ln(abs(x^2 - 1)) - 1/4*ln(abs(x/(sqrt(2) -
sqrt(-x^2 + 2)) - (sqrt(2) - sqrt(-x^2 + 2))/x + 2)) + 1/4*ln(abs(x/(sqrt(2) -
sqrt(-x^2 + 2)) - (sqrt(2) - sqrt(-x^2 + 2))/x - 2))