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