Optimal. Leaf size=37 \[ \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {280, 330, 52} \begin {gather*} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 280
Rule 330
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx &=\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{2} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 72, normalized size = 1.95 \begin {gather*} \frac {\sqrt {\sqrt {x}+1} \sqrt {x} \left (\sqrt {x}-1\right )+2 \sqrt {1-\sqrt {x}} \sin ^{-1}\left (\frac {\sqrt {1-\sqrt {x}}}{\sqrt {2}}\right )}{\sqrt {\sqrt {x}-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.17, size = 268, normalized size = 7.24 \begin {gather*} \frac {-4 \sqrt {\sqrt {x}+1} \left (-20 x^{3/2}-4 x+96 \sqrt {x}+48\right )-4 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \left (-7 x^{3/2}-28 x+10 \sqrt {x}+84\right )+\sqrt {3} \left (-4 \sqrt {\sqrt {x}-1} \left (12 x^{3/2}+20 x-32 \sqrt {x}-48\right )-4 \left (14 x^{3/2}+4 x^2-18 x-70 \sqrt {x}-28\right )\right )}{\sqrt {3} \sqrt {\sqrt {x}+1} \left (-48 \sqrt {x}-32\right )+\sqrt {\sqrt {x}-1} \left (\sqrt {3} \sqrt {\sqrt {x}+1} \left (-16 \sqrt {x}-56\right )+80 \sqrt {x}+96\right )+28 x+112 \sqrt {x}+56}+4 \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 [A] time = 0.41, size = 46, normalized size = 1.24 \begin {gather*} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{2} \, \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 = 0.18, size = 57, normalized size = 1.54 \begin {gather*} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} {\left (\sqrt {x} - 2\right )} + 2 \, \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + 2 \, \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 = 72, normalized size = 1.95 \begin {gather*} -\frac {\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}}+\sqrt {\sqrt {x}-1}\, \left (\sqrt {x}+1\right )^{\frac {3}{2}}-\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 [A] time = 0.49, size = 26, normalized size = 0.70 \begin {gather*} \sqrt {x - 1} \sqrt {x} - \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 = 5.07, size = 41, normalized size = 1.11 \begin {gather*} \sqrt {x}\,\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}-\ln \left (\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}+\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 {\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}{\sqrt {x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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