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 = 0.0118503, 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 \[ -\sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 - x]*Sqrt[x]),x]

[Out]

-ArcSin[1 - 2*x]

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Rubi in Sympy [A]  time = 2.40306, size = 5, normalized size = 0.62 \[ \operatorname{asin}{\left (2 x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(1/2)/x**(1/2),x)

[Out]

asin(2*x - 1)

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Mathematica [B]  time = 0.0144386, size = 38, normalized size = 4.75 \[ \frac{2 \sqrt{x-1} \sqrt{x} \log \left (\sqrt{x-1}+\sqrt{x}\right )}{\sqrt{-(x-1) x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[1 - x]*Sqrt[x]),x]

[Out]

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

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Maple [B]  time = 0.017, size = 27, normalized size = 3.4 \[{\arcsin \left ( -1+2\,x \right ) \sqrt{x \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{x}}}} \]

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(-1+2*x)

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Maxima [A]  time = 1.50174, size = 19, normalized size = 2.38 \[ -2 \, \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x)*sqrt(-x + 1)),x, algorithm="maxima")

[Out]

-2*arctan(sqrt(-x + 1)/sqrt(x))

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Fricas [A]  time = 0.2262, size = 19, normalized size = 2.38 \[ -2 \, \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x)*sqrt(-x + 1)),x, algorithm="fricas")

[Out]

-2*arctan(sqrt(-x + 1)/sqrt(x))

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Sympy [A]  time = 3.55382, size = 20, normalized size = 2.5 \[ \begin{cases} - 2 i \operatorname{acosh}{\left (\sqrt{x} \right )} & \text{for}\: \left |{x}\right | > 1 \\2 \operatorname{asin}{\left (\sqrt{x} \right )} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.206095, size = 8, normalized size = 1. \[ 2 \, \arcsin \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x)*sqrt(-x + 1)),x, algorithm="giac")

[Out]

2*arcsin(sqrt(x))