3.981 \(\int \frac{1}{\sqrt{(1-x) x}} \, dx\)

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

[Out]

-ArcSin[1 - 2*x]

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Rubi [A]  time = 0.0049172, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1979, 619, 216} \[ -\sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(1 - x)*x],x]

[Out]

-ArcSin[1 - 2*x]

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

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]

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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{(1-x) x}} \, dx &=\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*}

Mathematica [A]  time = 0.0076131, size = 12, normalized size = 1.5 \[ -2 \sin ^{-1}\left (\sqrt{1-x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(1 - x)*x],x]

[Out]

-2*ArcSin[Sqrt[1 - x]]

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Maple [A]  time = 0.003, size = 7, normalized size = 0.9 \begin{align*} \arcsin \left ( 2\,x-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1-x)*x)^(1/2),x)

[Out]

arcsin(2*x-1)

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Maxima [A]  time = 1.54528, size = 8, normalized size = 1. \begin{align*} \arcsin \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(2*x - 1)

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Fricas [B]  time = 1.4491, size = 39, normalized size = 4.88 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x^{2} + x}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(-x^2 + x)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(x*(1 - x)), x)

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Giac [A]  time = 1.20982, size = 8, normalized size = 1. \begin{align*} \arcsin \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(2*x - 1)