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

Optimal. Leaf size=20 \[ -\sinh ^{-1}\left (\frac{1-2 \sqrt{x-1}}{\sqrt{3}}\right ) \]

[Out]

-ArcSinh[(1 - 2*Sqrt[-1 + x])/Sqrt[3]]

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Rubi [A]  time = 0.179992, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\sinh ^{-1}\left (\frac{1-2 \sqrt{x-1}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcSinh[(1 - 2*Sqrt[-1 + x])/Sqrt[3]]

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Rubi in Sympy [A]  time = 7.18844, size = 24, normalized size = 1.2 \[ \operatorname{atanh}{\left (\frac{2 \sqrt{x - 1} - 1}{2 \sqrt{x - \sqrt{x - 1}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh((2*sqrt(x - 1) - 1)/(2*sqrt(x - sqrt(x - 1))))

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Mathematica [A]  time = 0.0218932, size = 18, normalized size = 0.9 \[ \sinh ^{-1}\left (\frac{2 \sqrt{x-1}-1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcSinh[(-1 + 2*Sqrt[-1 + x])/Sqrt[3]]

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Maple [A]  time = 0.009, size = 14, normalized size = 0.7 \[{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( \sqrt{-1+x}-{\frac{1}{2}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsinh(2/3*3^(1/2)*((-1+x)^(1/2)-1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.483548, size = 50, normalized size = 2.5 \[ \frac{1}{2} \, \log \left (4 \, \sqrt{x - \sqrt{x - 1}}{\left (2 \, \sqrt{x - 1} - 1\right )} + 8 \, x - 8 \, \sqrt{x - 1} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(4*sqrt(x - sqrt(x - 1))*(2*sqrt(x - 1) - 1) + 8*x - 8*sqrt(x - 1) - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.262849, size = 34, normalized size = 1.7 \[ -{\rm ln}\left (2 \, \sqrt{x - \sqrt{x - 1}} - 2 \, \sqrt{x - 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(2*sqrt(x - sqrt(x - 1)) - 2*sqrt(x - 1) + 1)