Optimal. Leaf size=20 \[ \log (1-x)+2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6098, 31} \[ \log (1-x)+2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 6098
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \, dx &=2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right )-\int \frac {1}{1-x} \, dx\\ &=2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right )+\log (1-x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ \log (1-x)+2 \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 24, normalized size = 1.20 \[ \sqrt {x} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (\sqrt {x}\right )}{\sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 15, normalized size = 0.75 \[ 2 \,\mathrm {arccoth}\left (\sqrt {x}\right ) \sqrt {x}+\ln \left (-1+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 16, normalized size = 0.80 \[ 2 \, \sqrt {x} \operatorname {arcoth}\left (\sqrt {x}\right ) + \log \left (-x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 14, normalized size = 0.70 \[ \ln \left (x-1\right )+2\,\sqrt {x}\,\mathrm {acoth}\left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.54, size = 87, normalized size = 4.35 \[ \frac {2 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x - 1} - \frac {2 \sqrt {x} \operatorname {acoth}{\left (\sqrt {x} \right )}}{x - 1} + \frac {2 x \log {\left (\sqrt {x} + 1 \right )}}{x - 1} - \frac {2 x \operatorname {acoth}{\left (\sqrt {x} \right )}}{x - 1} - \frac {2 \log {\left (\sqrt {x} + 1 \right )}}{x - 1} + \frac {2 \operatorname {acoth}{\left (\sqrt {x} \right )}}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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