Optimal. Leaf size=20 \[ \log (1-x)+2 \sqrt{x} \coth ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0092563, antiderivative size = 20, normalized size of antiderivative = 1., 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 6098
Rule 31
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*}
Mathematica [A] time = 0.007357, size = 20, normalized size = 1. \[ \log (1-x)+2 \sqrt{x} \coth ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 15, normalized size = 0.8 \begin{align*} 2\,{\rm arccoth} \left (\sqrt{x}\right )\sqrt{x}+\ln \left ( -1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967133, size = 22, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x} \operatorname{arcoth}\left (\sqrt{x}\right ) + \log \left (-x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64507, size = 74, normalized size = 3.7 \begin{align*} \sqrt{x} \log \left (\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\right ) + \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.780662, size = 87, normalized size = 4.35 \begin{align*} \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} \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 \frac{\operatorname{arcoth}\left (\sqrt{x}\right )}{\sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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