Optimal. Leaf size=20 \[ \log (1-x)+2 \sqrt{x} \tanh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0091329, 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 = {6097, 31} \[ \log (1-x)+2 \sqrt{x} \tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6097
Rule 31
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \sqrt{x} \tanh ^{-1}\left (\sqrt{x}\right )-\int \frac{1}{1-x} \, dx\\ &=2 \sqrt{x} \tanh ^{-1}\left (\sqrt{x}\right )+\log (1-x)\\ \end{align*}
Mathematica [A] time = 0.0084416, size = 20, normalized size = 1. \[ \log (1-x)+2 \sqrt{x} \tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 17, normalized size = 0.9 \begin{align*} \ln \left ( 1-x \right ) +2\,{\it Artanh} \left ( \sqrt{x} \right ) \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954426, size = 22, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x} \operatorname{artanh}\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 = 1.70806, size = 76, normalized size = 3.8 \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 = 0.892376, size = 87, normalized size = 4.35 \begin{align*} \frac{2 x^{\frac{3}{2}} \operatorname{atanh}{\left (\sqrt{x} \right )}}{x - 1} - \frac{2 \sqrt{x} \operatorname{atanh}{\left (\sqrt{x} \right )}}{x - 1} + \frac{2 x \log{\left (\sqrt{x} + 1 \right )}}{x - 1} - \frac{2 x \operatorname{atanh}{\left (\sqrt{x} \right )}}{x - 1} - \frac{2 \log{\left (\sqrt{x} + 1 \right )}}{x - 1} + \frac{2 \operatorname{atanh}{\left (\sqrt{x} \right )}}{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18187, size = 34, normalized size = 1.7 \begin{align*} \sqrt{x} \log \left (-\frac{\sqrt{x} + 1}{\sqrt{x} - 1}\right ) + \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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