Optimal. Leaf size=31 \[ \frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {x}{3}+\frac {1}{3} \log (1-x) \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6098, 43} \[ \frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {x}{3}+\frac {1}{3} \log (1-x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 6098
Rubi steps
\begin {align*} \int \sqrt {x} \coth ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x}{1-x} \, dx\\ &=\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \left (-1+\frac {1}{1-x}\right ) \, dx\\ &=\frac {x}{3}+\frac {2}{3} x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} \log (1-x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.81 \[ \frac {1}{3} \left (2 x^{3/2} \coth ^{-1}\left (\sqrt {x}\right )+x+\log (1-x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 30, normalized size = 0.97 \[ \frac {1}{3} \, x^{\frac {3}{2}} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x - 1\right ) \]
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 \[ \int \sqrt {x} \operatorname {arcoth}\left (\sqrt {x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 0.97 \[ \frac {2 x^{\frac {3}{2}} \mathrm {arccoth}\left (\sqrt {x}\right )}{3}+\frac {x}{3}+\frac {\ln \left (-1+\sqrt {x}\right )}{3}+\frac {\ln \left (1+\sqrt {x}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 19, normalized size = 0.61 \[ \frac {2}{3} \, x^{\frac {3}{2}} \operatorname {arcoth}\left (\sqrt {x}\right ) + \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x - 1\right ) \]
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 \[ \int \sqrt {x}\,\mathrm {acoth}\left (\sqrt {x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.07, size = 39, normalized size = 1.26 \[ \frac {2 x^{\frac {3}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{3} + \frac {x}{3} + \frac {2 \log {\left (\sqrt {x} + 1 \right )}}{3} - \frac {2 \operatorname {acoth}{\left (\sqrt {x} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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