3.310 \(\int \frac{\left (2 x-x^2\right )^{3/2}}{2-2 x} \, dx\)

Optimal. Leaf size=53 \[ -\frac{1}{6} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]

[Out]

-Sqrt[2*x - x^2]/2 - (2*x - x^2)^(3/2)/6 + ArcTanh[Sqrt[2*x - x^2]]/2

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Rubi [A]  time = 0.0775751, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{1}{6} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2*x - x^2)^(3/2)/(2 - 2*x),x]

[Out]

-Sqrt[2*x - x^2]/2 - (2*x - x^2)^(3/2)/6 + ArcTanh[Sqrt[2*x - x^2]]/2

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Rubi in Sympy [A]  time = 13.823, size = 36, normalized size = 0.68 \[ - \frac{\left (- x^{2} + 2 x\right )^{\frac{3}{2}}}{6} - \frac{\sqrt{- x^{2} + 2 x}}{2} + \frac{\operatorname{atanh}{\left (\sqrt{- x^{2} + 2 x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+2*x)**(3/2)/(2-2*x),x)

[Out]

-(-x**2 + 2*x)**(3/2)/6 - sqrt(-x**2 + 2*x)/2 + atanh(sqrt(-x**2 + 2*x))/2

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Mathematica [A]  time = 0.0855779, size = 84, normalized size = 1.58 \[ \frac{\sqrt{-(x-2) x} \left (\sqrt{x-2} \sqrt{x} \left (x^2-2 x-3\right )+3 \tan ^{-1}\left (\frac{\sqrt{x}-2}{\sqrt{x-2}}\right )+3 \tan ^{-1}\left (\frac{\sqrt{x}+2}{\sqrt{x-2}}\right )\right )}{6 \sqrt{x-2} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2*x - x^2)^(3/2)/(2 - 2*x),x]

[Out]

(Sqrt[-((-2 + x)*x)]*(Sqrt[-2 + x]*Sqrt[x]*(-3 - 2*x + x^2) + 3*ArcTan[(-2 + Sqr
t[x])/Sqrt[-2 + x]] + 3*ArcTan[(2 + Sqrt[x])/Sqrt[-2 + x]]))/(6*Sqrt[-2 + x]*Sqr
t[x])

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Maple [A]  time = 0.008, size = 42, normalized size = 0.8 \[ -{\frac{1}{6} \left ( - \left ( -1+x \right ) ^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2}\sqrt{- \left ( -1+x \right ) ^{2}+1}}+{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+2*x)^(3/2)/(2-2*x),x)

[Out]

-1/6*(-(-1+x)^2+1)^(3/2)-1/2*(-(-1+x)^2+1)^(1/2)+1/2*arctanh(1/(-(-1+x)^2+1)^(1/
2))

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Maxima [A]  time = 0.75893, size = 78, normalized size = 1.47 \[ -\frac{1}{6} \,{\left (-x^{2} + 2 \, x\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac{2}{{\left | x - 1 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(-x^2 + 2*x)^(3/2)/(x - 1),x, algorithm="maxima")

[Out]

-1/6*(-x^2 + 2*x)^(3/2) - 1/2*sqrt(-x^2 + 2*x) + 1/2*log(2*sqrt(-x^2 + 2*x)/abs(
x - 1) + 2/abs(x - 1))

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Fricas [A]  time = 0.242045, size = 88, normalized size = 1.66 \[ \frac{1}{6} \,{\left (x^{2} - 2 \, x - 3\right )} \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - \frac{1}{2} \, \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(-x^2 + 2*x)^(3/2)/(x - 1),x, algorithm="fricas")

[Out]

1/6*(x^2 - 2*x - 3)*sqrt(-x^2 + 2*x) + 1/2*log((x + sqrt(-x^2 + 2*x))/x) - 1/2*l
og(-(x - sqrt(-x^2 + 2*x))/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+2*x)**(3/2)/(2-2*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217275, size = 63, normalized size = 1.19 \[ \frac{1}{6} \,{\left ({\left (x - 2\right )} x - 3\right )} \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \,{\rm ln}\left (-\frac{2 \,{\left (\sqrt{-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(-x^2 + 2*x)^(3/2)/(x - 1),x, algorithm="giac")

[Out]

1/6*((x - 2)*x - 3)*sqrt(-x^2 + 2*x) - 1/2*ln(-2*(sqrt(-x^2 + 2*x) - 1)/abs(-2*x
+ 2))