Optimal. Leaf size=20 \[ -2 \sinh ^{-1}\left (\frac {1-2 \sqrt {x-1}}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {619, 215} \[ -2 \sinh ^{-1}\left (\frac {1-2 \sqrt {x-1}}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 619
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x} \sqrt {-\sqrt {-1+x}+x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x+x^2}} \, dx,x,\sqrt {-1+x}\right )\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,-1+2 \sqrt {-1+x}\right )}{\sqrt {3}}\\ &=-2 \sinh ^{-1}\left (\frac {1-2 \sqrt {-1+x}}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \[ 2 \sinh ^{-1}\left (\frac {2 \sqrt {x-1}-1}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 35, normalized size = 1.75 \[ \log \left (4 \, \sqrt {x - \sqrt {x - 1}} {\left (2 \, \sqrt {x - 1} - 1\right )} + 8 \, x - 8 \, \sqrt {x - 1} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 25, normalized size = 1.25 \[ -2 \, \log \left (2 \, \sqrt {x - \sqrt {x - 1}} - 2 \, \sqrt {x - 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.80 \[ 2 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {x -1}-\frac {1}{2}\right )}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x - \sqrt {x - 1}} \sqrt {x - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{\sqrt {x-\sqrt {x-1}}\,\sqrt {x-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {x - \sqrt {x - 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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