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 = 0.0135038, antiderivative size = 31, normalized size of antiderivative = 1., 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 6098
Rule 43
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*}
Mathematica [A] time = 0.0101959, 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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Maple [A] time = 0.036, size = 30, normalized size = 1. \begin{align*}{\frac{2}{3}{x}^{{\frac{3}{2}}}{\rm arccoth} \left (\sqrt{x}\right )}+{\frac{x}{3}}+{\frac{1}{3}\ln \left ( -1+\sqrt{x} \right ) }+{\frac{1}{3}\ln \left ( 1+\sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990299, size = 26, normalized size = 0.84 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65532, size = 96, normalized size = 3.1 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.60655, size = 39, normalized size = 1.26 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \operatorname{arcoth}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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