3.1162 \(\int \frac{1}{\sqrt{3-x} \sqrt{-2+x}} \, dx\)

Optimal. Leaf size=8 \[ -\sin ^{-1}(5-2 x) \]

[Out]

-ArcSin[5 - 2*x]

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Rubi [A]  time = 0.0159681, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\sin ^{-1}(5-2 x) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[3 - x]*Sqrt[-2 + x]),x]

[Out]

-ArcSin[5 - 2*x]

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Rubi in Sympy [A]  time = 3.08624, size = 20, normalized size = 2.5 \[ - \operatorname{atan}{\left (\frac{- 2 x + 5}{2 \sqrt{- x^{2} + 5 x - 6}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-x)**(1/2)/(-2+x)**(1/2),x)

[Out]

-atan((-2*x + 5)/(2*sqrt(-x**2 + 5*x - 6)))

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Mathematica [B]  time = 0.0137138, size = 36, normalized size = 4.5 \[ \frac{2 \sqrt{x-3} \sqrt{x-2} \sinh ^{-1}\left (\sqrt{x-3}\right )}{\sqrt{-(x-3) (x-2)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[3 - x]*Sqrt[-2 + x]),x]

[Out]

(2*Sqrt[-3 + x]*Sqrt[-2 + x]*ArcSinh[Sqrt[-3 + x]])/Sqrt[-((-3 + x)*(-2 + x))]

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Maple [B]  time = 0.006, size = 31, normalized size = 3.9 \[{\arcsin \left ( -5+2\,x \right ) \sqrt{ \left ( -2+x \right ) \left ( 3-x \right ) }{\frac{1}{\sqrt{3-x}}}{\frac{1}{\sqrt{-2+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-x)^(1/2)/(-2+x)^(1/2),x)

[Out]

((-2+x)*(3-x))^(1/2)/(-2+x)^(1/2)/(3-x)^(1/2)*arcsin(-5+2*x)

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Maxima [A]  time = 1.47485, size = 8, normalized size = 1. \[ \arcsin \left (2 \, x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 2)*sqrt(-x + 3)),x, algorithm="maxima")

[Out]

arcsin(2*x - 5)

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Fricas [A]  time = 0.209715, size = 27, normalized size = 3.38 \[ \arctan \left (\frac{2 \, x - 5}{2 \, \sqrt{x - 2} \sqrt{-x + 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 2)*sqrt(-x + 3)),x, algorithm="fricas")

[Out]

arctan(1/2*(2*x - 5)/(sqrt(x - 2)*sqrt(-x + 3)))

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Sympy [A]  time = 5.07244, size = 26, normalized size = 3.25 \[ \begin{cases} - 2 i \operatorname{acosh}{\left (\sqrt{x - 2} \right )} & \text{for}\: \left |{x - 2}\right | > 1 \\2 \operatorname{asin}{\left (\sqrt{x - 2} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3-x)**(1/2)/(-2+x)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220987, size = 11, normalized size = 1.38 \[ 2 \, \arcsin \left (\sqrt{x - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 2)*sqrt(-x + 3)),x, algorithm="giac")

[Out]

2*arcsin(sqrt(x - 2))