3.98 \(\int \frac{1+x}{\sqrt{2 x-x^2}} \, dx\)

Optimal. Leaf size=24 \[ -\sqrt{2 x-x^2}-2 \sin ^{-1}(1-x) \]

[Out]

-Sqrt[2*x - x^2] - 2*ArcSin[1 - x]

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Rubi [A]  time = 0.024367, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\sqrt{2 x-x^2}-2 \sin ^{-1}(1-x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/Sqrt[2*x - x^2],x]

[Out]

-Sqrt[2*x - x^2] - 2*ArcSin[1 - x]

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Rubi in Sympy [A]  time = 1.90138, size = 15, normalized size = 0.62 \[ - \sqrt{- x^{2} + 2 x} + 2 \operatorname{asin}{\left (x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 2*x) + 2*asin(x - 1)

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Mathematica [A]  time = 0.0270264, size = 45, normalized size = 1.88 \[ \frac{(x-2) x+4 \sqrt{x-2} \sqrt{x} \log \left (\sqrt{x-2}+\sqrt{x}\right )}{\sqrt{-(x-2) x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 21, normalized size = 0.9 \[ 2\,\arcsin \left ( -1+x \right ) -\sqrt{-{x}^{2}+2\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(-x^2+2*x)^(1/2),x)

[Out]

2*arcsin(-1+x)-(-x^2+2*x)^(1/2)

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Maxima [A]  time = 1.50315, size = 30, normalized size = 1.25 \[ -\sqrt{-x^{2} + 2 \, x} - 2 \, \arcsin \left (-x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 2*x) - 2*arcsin(-x + 1)

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Fricas [A]  time = 0.247878, size = 43, normalized size = 1.79 \[ -\sqrt{-x^{2} + 2 \, x} - 4 \, \arctan \left (\frac{\sqrt{-x^{2} + 2 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 2*x) - 4*arctan(sqrt(-x^2 + 2*x)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + 1)/sqrt(-x*(x - 2)), x)

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GIAC/XCAS [A]  time = 0.212486, size = 27, normalized size = 1.12 \[ -\sqrt{-x^{2} + 2 \, x} + 2 \, \arcsin \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 2*x) + 2*arcsin(x - 1)