Optimal. Leaf size=22 \[ \sqrt{x}-\tanh ^{-1}\left (\sqrt{x}\right )+x \coth ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.006253, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6092, 50, 63, 206} \[ \sqrt{x}-\tanh ^{-1}\left (\sqrt{x}\right )+x \coth ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6092
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \coth ^{-1}\left (\sqrt{x}\right ) \, dx &=x \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{\sqrt{x}}{1-x} \, dx\\ &=\sqrt{x}+x \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{1}{(1-x) \sqrt{x}} \, dx\\ &=\sqrt{x}+x \coth ^{-1}\left (\sqrt{x}\right )-\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )\\ &=\sqrt{x}+x \coth ^{-1}\left (\sqrt{x}\right )-\tanh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0059219, size = 22, normalized size = 1. \[ \sqrt{x}-\tanh ^{-1}\left (\sqrt{x}\right )+x \coth ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 27, normalized size = 1.2 \begin{align*} x{\rm arccoth} \left (\sqrt{x}\right )+\sqrt{x}+{\frac{1}{2}\ln \left ( -1+\sqrt{x} \right ) }-{\frac{1}{2}\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 = 0.951836, size = 35, normalized size = 1.59 \begin{align*} x \operatorname{arcoth}\left (\sqrt{x}\right ) + \sqrt{x} - \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63877, size = 76, normalized size = 3.45 \begin{align*} \frac{1}{2} \,{\left (x - 1\right )} \log \left (\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\right ) + \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acoth}{\left (\sqrt{x} \right )}\, dx \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 \operatorname{arcoth}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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