3.10.28 \(\int \frac {-1+x}{\sqrt {2 x-x^2}} \, dx\) [928]

Optimal. Leaf size=15 \[ -\sqrt {2 x-x^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {643} \begin {gather*} -\sqrt {2 x-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x)/Sqrt[2*x - x^2],x]

[Out]

-Sqrt[2*x - x^2]

Rule 643

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*((a + b*x + c*x^2)^(p +
1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {-1+x}{\sqrt {2 x-x^2}} \, dx &=-\sqrt {2 x-x^2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 12, normalized size = 0.80 \begin {gather*} -\sqrt {-((-2+x) x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x)/Sqrt[2*x - x^2],x]

[Out]

-Sqrt[-((-2 + x)*x)]

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Maple [A]
time = 0.22, size = 14, normalized size = 0.93

method	result	size
default	\(-\sqrt {-x^{2}+2 x}\)	\(14\)
trager	\(-\sqrt {-x^{2}+2 x}\)	\(14\)
risch	\(\frac {x \left (x -2\right )}{\sqrt {-x \left (x -2\right )}}\)	\(14\)
gosper	\(\frac {x \left (x -2\right )}{\sqrt {-x^{2}+2 x}}\)	\(17\)
meijerg	\(-2 \arcsin \left (\frac {\sqrt {2}\, \sqrt {x}}{2}\right )+\frac {2 i \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {1-\frac {x}{2}}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {2}\, \sqrt {x}}{2}\right )\right )}{\sqrt {\pi }}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+x)/(-x^2+2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 13, normalized size = 0.87 \begin {gather*} -\sqrt {-x^{2} + 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 2*x)

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Fricas [A]
time = 0.41, size = 13, normalized size = 0.87 \begin {gather*} -\sqrt {-x^{2} + 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 2*x)

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Sympy [A]
time = 0.05, size = 10, normalized size = 0.67 \begin {gather*} - \sqrt {- x^{2} + 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x**2 + 2*x)

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Giac [A]
time = 4.34, size = 13, normalized size = 0.87 \begin {gather*} -\sqrt {-x^{2} + 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 2*x)

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Mupad [B]
time = 3.64, size = 10, normalized size = 0.67 \begin {gather*} -\sqrt {-x\,\left (x-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/(2*x - x^2)^(1/2),x)

[Out]

-(-x*(x - 2))^(1/2)

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