3.1.56 \(\int \frac {1}{\sqrt {x-x^2}} \, dx\) [56]

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

[Out]

arcsin(-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 = {633, 222} \begin {gather*} -\text {ArcSin}(1-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[x - x^2],x]

[Out]

-ArcSin[1 - 2*x]

Rule 222

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 633

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 &=-\text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(38\) vs. \(2(8)=16\).
time = 0.03, size = 38, normalized size = 4.75 \begin {gather*} \frac {2 \sqrt {-1+x} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-1+x}}\right )}{\sqrt {-((-1+x) x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[x - x^2],x]

[Out]

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

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Maple [A]
time = 0.08, size = 7, normalized size = 0.88

method	result	size
default	\(\arcsin \left (2 x -1\right )\)	\(7\)
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(2*x-1)

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Maxima [A]
time = 2.44, size = 6, normalized size = 0.75 \begin {gather*} \arcsin \left (2 \, x - 1\right ) \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (6) = 12\).
time = 0.67, size = 16, normalized size = 2.00 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x^{2} + x}}{x}\right ) \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- x^{2} + x}}\, dx \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (6) = 12\).
time = 0.69, size = 25, normalized size = 3.12 \begin {gather*} \frac {1}{4} \, \sqrt {-x^{2} + x} {\left (2 \, x - 1\right )} + \frac {1}{8} \, \arcsin \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(-x^2 + x)*(2*x - 1) + 1/8*arcsin(2*x - 1)

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

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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