Optimal. Leaf size=8 \[ 2 \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {330, 52} \begin {gather*} 2 \cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 330
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=2 \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [B] time = 0.01, size = 26, normalized size = 3.25 \begin {gather*} 4 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x}-1}}{\sqrt {\sqrt {x}+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.02, size = 38, normalized size = 4.75 \begin {gather*} -8 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x}-1}-1}{\sqrt {3}-\sqrt {\sqrt {x}+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 27, normalized size = 3.38 \begin {gather*} -\log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 20, normalized size = 2.50 \begin {gather*} -4 \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 40, normalized size = 5.00 \begin {gather*} \frac {2 \sqrt {\left (\sqrt {x}-1\right ) \left (\sqrt {x}+1\right )}\, \ln \left (\sqrt {x}+\sqrt {x -1}\right )}{\sqrt {\sqrt {x}+1}\, \sqrt {\sqrt {x}-1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 16, normalized size = 2.00 \begin {gather*} 2 \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.29, size = 6, normalized size = 0.75 \begin {gather*} 2\,\mathrm {acosh}\left (\sqrt {x}\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {x} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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