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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.156734, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -2 \sinh ^{-1}\left (\frac{1-2 \sqrt{x-1}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.05316, size = 26, normalized size = 1.3 \[ 2 \operatorname{atanh}{\left (\frac{2 \sqrt{x - 1} - 1}{2 \sqrt{x - \sqrt{x - 1}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0160907, size = 20, normalized size = 1. \[ 2 \sinh ^{-1}\left (\frac{2 \sqrt{x-1}-1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 16, normalized size = 0.8 \[ 2\,{\it Arcsinh} \left ( 2/3\,\sqrt{3} \left ( \sqrt{-1+x}-1/2 \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.476913, size = 47, normalized size = 2.35 \[ \log \left (4 \, \sqrt{x - \sqrt{x - 1}}{\left (2 \, \sqrt{x - 1} - 1\right )} + 8 \, x - 8 \, \sqrt{x - 1} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286016, size = 34, normalized size = 1.7 \[ -2 \,{\rm ln}\left (2 \, \sqrt{x - \sqrt{x - 1}} - 2 \, \sqrt{x - 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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