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

Optimal. Leaf size=8 \[ -\sin ^{-1}(1-2 x) \]

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {619, 216} \[ -\sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[x - x^2],x]

[Out]

-ArcSin[1 - 2*x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x-x^2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right )\\ &=-\sin ^{-1}(1-2 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.50 \[ -2 \sin ^{-1}\left (\sqrt {1-x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[x - x^2],x]

[Out]

-2*ArcSin[Sqrt[1 - x]]

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IntegrateAlgebraic [B]  time = 0.08, size = 18, normalized size = 2.25 \[ -2 \tan ^{-1}\left (\frac {\sqrt {x-x^2}}{x}\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[x - x^2],x]

[Out]

-2*ArcTan[Sqrt[x - x^2]/x]

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fricas [B]  time = 0.93, size = 16, normalized size = 2.00 \[ -2 \, \arctan \left (\frac {\sqrt {-x^{2} + x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(-x^2 + x)/x)

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giac [A]  time = 0.86, size = 6, normalized size = 0.75 \[ \arcsin \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(2*x - 1)

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maple [A]  time = 0.32, size = 7, normalized size = 0.88

method	result	size
default	\(\arcsin \left (-1+2 x \right )\)	\(7\)
meijerg	\(2 \arcsin \left (\sqrt {x}\right )\)	\(7\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {-x^{2}+x}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(-1+2*x)

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maxima [A]  time = 0.97, size = 6, normalized size = 0.75 \[ \arcsin \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(2*x - 1)

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mupad [B]  time = 0.16, size = 6, normalized size = 0.75 \[ \mathrm {asin}\left (2\,x-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - x^2)^(1/2),x)

[Out]

asin(2*x - 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- x^{2} + x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-x**2 + x), x)

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