Optimal. Leaf size=24 \[ -\log (1-x)+\log (x)-\frac{2 \coth ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
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Rubi [A] time = 0.0104366, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6098, 36, 31, 29} \[ -\log (1-x)+\log (x)-\frac{2 \coth ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 6098
Rule 36
Rule 31
Rule 29
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \, dx &=-\frac{2 \coth ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}+\int \frac{1}{(1-x) x} \, dx\\ &=-\frac{2 \coth ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}+\int \frac{1}{1-x} \, dx+\int \frac{1}{x} \, dx\\ &=-\frac{2 \coth ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}-\log (1-x)+\log (x)\\ \end{align*}
Mathematica [A] time = 0.0164811, size = 24, normalized size = 1. \[ -\log (1-x)+\log (x)-\frac{2 \coth ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 29, normalized size = 1.2 \begin{align*} -2\,{\frac{{\rm arccoth} \left (\sqrt{x}\right )}{\sqrt{x}}}-\ln \left ( -1+\sqrt{x} \right ) +\ln \left ( x \right ) -\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 = 0.986294, size = 24, normalized size = 1. \begin{align*} -\frac{2 \, \operatorname{arcoth}\left (\sqrt{x}\right )}{\sqrt{x}} - \log \left (x - 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63988, size = 99, normalized size = 4.12 \begin{align*} -\frac{x \log \left (x - 1\right ) - x \log \left (x\right ) + \sqrt{x} \log \left (\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.15684, size = 126, normalized size = 5.25 \begin{align*} - \frac{2 x^{\frac{3}{2}} \operatorname{acoth}{\left (\sqrt{x} \right )}}{x^{2} - x} + \frac{2 \sqrt{x} \operatorname{acoth}{\left (\sqrt{x} \right )}}{x^{2} - x} + \frac{x^{2} \log{\left (x \right )}}{x^{2} - x} - \frac{2 x^{2} \log{\left (\sqrt{x} + 1 \right )}}{x^{2} - x} + \frac{2 x^{2} \operatorname{acoth}{\left (\sqrt{x} \right )}}{x^{2} - x} - \frac{x \log{\left (x \right )}}{x^{2} - x} + \frac{2 x \log{\left (\sqrt{x} + 1 \right )}}{x^{2} - x} - \frac{2 x \operatorname{acoth}{\left (\sqrt{x} \right )}}{x^{2} - x} \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 )}{x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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