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 = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 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]

Rule 1979

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

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*}

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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[(1 - x)*x],x]

[Out]

-2*ArcSin[Sqrt[1 - x]]

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fricas [B]  time = 0.45, 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/((1-x)*x)^(1/2),x, algorithm="fricas")

[Out]

-2*arctan(sqrt(-x^2 + x)/x)

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giac [A]  time = 0.39, 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/((1-x)*x)^(1/2),x, algorithm="giac")

[Out]

arcsin(2*x - 1)

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maple [A]  time = 0.01, size = 7, normalized size = 0.88 \[ \arcsin \left (2 x -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(2*x-1)

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maxima [A]  time = 0.96, 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/((1-x)*x)^(1/2),x, algorithm="maxima")

[Out]

arcsin(2*x - 1)

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mupad [B]  time = 0.01, 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 - 1))^(1/2),x)

[Out]

asin(2*x - 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x \left (1 - x\right )}}\, dx \]

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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