Optimal. Leaf size=35 \[ \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+\cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 35, 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 = {323, 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 323
Rule 330
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\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 = 0.01, size = 55, normalized size = 1.57 \begin {gather*} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+2 \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 = 1.27, size = 268, normalized size = 7.66 \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 = 1.31 \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 [A] time = 0.22, size = 39, normalized size = 1.11 \begin {gather*} \sqrt {x} \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 [A] time = 0.06, size = 41, normalized size = 1.17 \begin {gather*} \frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (\ln \left (\sqrt {x}+\sqrt {x -1}\right )+\sqrt {x -1}\, \sqrt {x}\right )}{\sqrt {x -1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 24, normalized size = 0.69 \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 [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 27.91, size = 83, normalized size = 2.37 \begin {gather*} \frac {{G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & - \frac {1}{2}, - \frac {1}{2}, 0, 1 \\-1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 0 & \end {matrix} \middle | {\frac {1}{x}} \right )}}{2 \pi ^{\frac {3}{2}}} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & - \frac {3}{2}, -1, -1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x}} \right )}}{2 \pi ^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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