Optimal. Leaf size=20 \[ -2 \sinh ^{-1}\left (\frac{1-2 \sqrt{x-1}}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.0778544, 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, 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 619
Rule 215
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*}
Mathematica [A] time = 0.0206398, size = 20, normalized size = 1. \[ 2 \sinh ^{-1}\left (\frac{2 \sqrt{x-1}-1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 16, normalized size = 0.8 \begin{align*} 2\,{\it Arcsinh} \left ( 2/3\,\sqrt{3} \left ( \sqrt{x-1}-1/2 \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - \sqrt{x - 1}} \sqrt{x - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.70194, size = 103, normalized size = 5.15 \begin{align*} \log \left (4 \, \sqrt{x - \sqrt{x - 1}}{\left (2 \, \sqrt{x - 1} - 1\right )} + 8 \, x - 8 \, \sqrt{x - 1} - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - 1} \sqrt{x - \sqrt{x - 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22316, size = 34, normalized size = 1.7 \begin{align*} -2 \, \log \left (2 \, \sqrt{x - \sqrt{x - 1}} - 2 \, \sqrt{x - 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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