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

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

[Out]

-ArcSin[1 - 2*x]

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Rubi [A]  time = 0.0030581, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {53, 619, 216} \[ -\sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x]*Sqrt[x]),x]

[Out]

-ArcSin[1 - 2*x]

Rule 53

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

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} \sqrt{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 = 0.0079737, size = 12, normalized size = 1.5 \[ -2 \sin ^{-1}\left (\sqrt{1-x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - x]*Sqrt[x]),x]

[Out]

-2*ArcSin[Sqrt[1 - x]]

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Maple [B]  time = 0.003, size = 27, normalized size = 3.4 \begin{align*}{\arcsin \left ( 2\,x-1 \right ) \sqrt{x \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.55005, size = 19, normalized size = 2.38 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(-x + 1)/sqrt(x))

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Fricas [B]  time = 1.53014, size = 45, normalized size = 5.62 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(-x + 1)/sqrt(x))

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Sympy [A]  time = 1.00491, size = 20, normalized size = 2.5 \begin{align*} \begin{cases} - 2 i \operatorname{acosh}{\left (\sqrt{x} \right )} & \text{for}\: \left |{x}\right | > 1 \\2 \operatorname{asin}{\left (\sqrt{x} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*acosh(sqrt(x)), Abs(x) > 1), (2*asin(sqrt(x)), True))

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Giac [A]  time = 1.06984, size = 8, normalized size = 1. \begin{align*} 2 \, \arcsin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arcsin(sqrt(x))