Optimal. Leaf size=19 \[ -\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {63, 207} \[ -\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (-1+2 x)} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \[ -\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 28, normalized size = 1.47 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} \sqrt {x} - 2 \, x - 1}{2 \, x - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.27, size = 32, normalized size = 1.68 \[ -\frac {1}{2} \, \sqrt {2} \log \left (\frac {1}{2} \, \sqrt {2} + \sqrt {x}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sqrt {x} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.74 \[ -\sqrt {2}\, \arctanh \left (\sqrt {2}\, \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.97, size = 28, normalized size = 1.47 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - 2 \, \sqrt {x}}{\sqrt {2} + 2 \, \sqrt {x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 13, normalized size = 0.68 \[ -\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2\,x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.31, size = 39, normalized size = 2.05 \[ \frac {\sqrt {2} \log {\left (\sqrt {x} - \frac {\sqrt {2}}{2} \right )}}{2} - \frac {\sqrt {2} \log {\left (\sqrt {x} + \frac {\sqrt {2}}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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