3.314 \(\int \frac{1}{(2-2 x) \left (2 x-x^2\right )^{5/2}} \, dx\)

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

[Out]

-1/(6*(2*x - x^2)^(3/2)) - 1/(2*Sqrt[2*x - x^2]) + ArcTanh[Sqrt[2*x - x^2]]/2

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Rubi [A]  time = 0.0787405, 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 \[ -\frac{1}{2 \sqrt{2 x-x^2}}-\frac{1}{6 \left (2 x-x^2\right )^{3/2}}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((2 - 2*x)*(2*x - x^2)^(5/2)),x]

[Out]

-1/(6*(2*x - x^2)^(3/2)) - 1/(2*Sqrt[2*x - x^2]) + ArcTanh[Sqrt[2*x - x^2]]/2

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Rubi in Sympy [A]  time = 14.0453, size = 39, normalized size = 0.74 \[ \frac{\operatorname{atanh}{\left (\sqrt{- x^{2} + 2 x} \right )}}{2} - \frac{1}{2 \sqrt{- x^{2} + 2 x}} - \frac{1}{6 \left (- x^{2} + 2 x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2-2*x)/(-x**2+2*x)**(5/2),x)

[Out]

atanh(sqrt(-x**2 + 2*x))/2 - 1/(2*sqrt(-x**2 + 2*x)) - 1/(6*(-x**2 + 2*x)**(3/2)
)

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Mathematica [A]  time = 0.117666, size = 84, normalized size = 1.58 \[ \frac{3 (x-2)^{3/2} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{x}-2}{\sqrt{x-2}}\right )+3 (x-2)^{3/2} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{x}+2}{\sqrt{x-2}}\right )+3 x^2-6 x-1}{6 (-(x-2) x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((2 - 2*x)*(2*x - x^2)^(5/2)),x]

[Out]

(-1 - 6*x + 3*x^2 + 3*(-2 + x)^(3/2)*x^(3/2)*ArcTan[(-2 + Sqrt[x])/Sqrt[-2 + x]]
+ 3*(-2 + x)^(3/2)*x^(3/2)*ArcTan[(2 + Sqrt[x])/Sqrt[-2 + x]])/(6*(-((-2 + x)*x
))^(3/2))

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Maple [A]  time = 0.007, size = 42, normalized size = 0.8 \[ -{\frac{1}{6} \left ( - \left ( -1+x \right ) ^{2}+1 \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{2}{\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}}}+{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2-2*x)/(-x^2+2*x)^(5/2),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 = 0.689887, size = 78, normalized size = 1.47 \[ -\frac{1}{2 \, \sqrt{-x^{2} + 2 \, x}} - \frac{1}{6 \,{\left (-x^{2} + 2 \, x\right )}^{\frac{3}{2}}} + \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac{2}{{\left | x - 1 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2/((-x^2 + 2*x)^(5/2)*(x - 1)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.221029, size = 150, normalized size = 2.83 \[ \frac{3 \,{\left (x^{2} - 2 \, x\right )} \sqrt{-x^{2} + 2 \, x} \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - 3 \,{\left (x^{2} - 2 \, x\right )} \sqrt{-x^{2} + 2 \, x} \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) - 3 \, x^{2} + 6 \, x + 1}{6 \,{\left (x^{2} - 2 \, x\right )} \sqrt{-x^{2} + 2 \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2/((-x^2 + 2*x)^(5/2)*(x - 1)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\int \frac{1}{x^{5} \sqrt{- x^{2} + 2 x} - 5 x^{4} \sqrt{- x^{2} + 2 x} + 8 x^{3} \sqrt{- x^{2} + 2 x} - 4 x^{2} \sqrt{- x^{2} + 2 x}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2-2*x)/(-x**2+2*x)**(5/2),x)

[Out]

-Integral(1/(x**5*sqrt(-x**2 + 2*x) - 5*x**4*sqrt(-x**2 + 2*x) + 8*x**3*sqrt(-x*
*2 + 2*x) - 4*x**2*sqrt(-x**2 + 2*x)), x)/2

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GIAC/XCAS [A]  time = 0.221023, size = 77, normalized size = 1.45 \[ \frac{{\left (3 \,{\left (x - 2\right )} x - 1\right )} \sqrt{-x^{2} + 2 \, x}}{6 \,{\left (x^{2} - 2 \, x\right )}^{2}} - \frac{1}{2} \,{\rm ln}\left (-\frac{2 \,{\left (\sqrt{-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2/((-x^2 + 2*x)^(5/2)*(x - 1)),x, algorithm="giac")

[Out]

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