3.2873 \(\int \frac{1}{\sqrt{-1+x} \sqrt{-12+8 x-x^2}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]

[Out]

(-2*EllipticF[ArcSin[Sqrt[6 - x]/2], 4/5])/Sqrt[5]

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Rubi [A]  time = 0.0573989, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 F\left (\sin ^{-1}\left (\frac{\sqrt{6-x}}{2}\right )|\frac{4}{5}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[-1 + x]*Sqrt[-12 + 8*x - x^2]),x]

[Out]

(-2*EllipticF[ArcSin[Sqrt[6 - x]/2], 4/5])/Sqrt[5]

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Rubi in Sympy [A]  time = 11.8463, size = 66, normalized size = 2.64 \[ - \frac{8 \sqrt{\frac{x}{5} - \frac{1}{5}} \sqrt{- \frac{x^{2}}{16} + \frac{x}{2} - \frac{3}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{- \frac{x}{2} + 3}}{2} \right )}\middle | \frac{4}{5}\right )}{\sqrt{x - 1} \sqrt{- x^{2} + 8 x - 12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-1+x)**(1/2)/(-x**2+8*x-12)**(1/2),x)

[Out]

-8*sqrt(x/5 - 1/5)*sqrt(-x**2/16 + x/2 - 3/4)*elliptic_f(asin(sqrt(2)*sqrt(-x/2
+ 3)/2), 4/5)/(sqrt(x - 1)*sqrt(-x**2 + 8*x - 12))

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Mathematica [B]  time = 0.099588, size = 68, normalized size = 2.72 \[ -\frac{2 \sqrt{\frac{x-6}{x-1}} \sqrt{\frac{x-2}{x-1}} (x-1) F\left (\sin ^{-1}\left (\frac{\sqrt{5}}{\sqrt{x-1}}\right )|\frac{1}{5}\right )}{\sqrt{5} \sqrt{-x^2+8 x-12}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[-1 + x]*Sqrt[-12 + 8*x - x^2]),x]

[Out]

(-2*Sqrt[(-6 + x)/(-1 + x)]*Sqrt[(-2 + x)/(-1 + x)]*(-1 + x)*EllipticF[ArcSin[Sq
rt[5]/Sqrt[-1 + x]], 1/5])/(Sqrt[5]*Sqrt[-12 + 8*x - x^2])

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Maple [B]  time = 0.02, size = 55, normalized size = 2.2 \[{\frac{2\,\sqrt{5}}{5\,{x}^{2}-40\,x+60}{\it EllipticF} \left ({\frac{1}{2}\sqrt{6-x}},{\frac{2\,\sqrt{5}}{5}} \right ) \sqrt{-2+x}\sqrt{6-x}\sqrt{-{x}^{2}+8\,x-12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-1+x)^(1/2)/(-x^2+8*x-12)^(1/2),x)

[Out]

2/5*EllipticF(1/2*(6-x)^(1/2),2/5*5^(1/2))*(-2+x)^(1/2)*(6-x)^(1/2)*(-x^2+8*x-12
)^(1/2)*5^(1/2)/(x^2-8*x+12)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x - 6\right ) \left (x - 2\right )} \sqrt{x - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-1+x)**(1/2)/(-x**2+8*x-12)**(1/2),x)

[Out]

Integral(1/(sqrt(-(x - 6)*(x - 2))*sqrt(x - 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{2} + 8 \, x - 12} \sqrt{x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^2 + 8*x - 12)*sqrt(x - 1)), x)