Optimal. Leaf size=42 \[ \frac{x^{3/2}}{6}+\frac{1}{2} x^2 \coth ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{2}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0107724, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6098, 50, 63, 206} \[ \frac{x^{3/2}}{6}+\frac{1}{2} x^2 \coth ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{2}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6098
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int x \coth ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{2} x^2 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{x^{3/2}}{1-x} \, dx\\ &=\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{\sqrt{x}}{1-x} \, dx\\ &=\frac{\sqrt{x}}{2}+\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{1}{(1-x) \sqrt{x}} \, dx\\ &=\frac{\sqrt{x}}{2}+\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{2}+\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \tanh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0145469, size = 52, normalized size = 1.24 \[ \frac{1}{12} \left (2 x^{3/2}+6 x^2 \coth ^{-1}\left (\sqrt{x}\right )+6 \sqrt{x}+3 \log \left (1-\sqrt{x}\right )-3 \log \left (\sqrt{x}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 37, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}{\rm arccoth} \left (\sqrt{x}\right )}+{\frac{1}{6}{x}^{{\frac{3}{2}}}}+{\frac{1}{2}\sqrt{x}}+{\frac{1}{4}\ln \left ( -1+\sqrt{x} \right ) }-{\frac{1}{4}\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 = 0.957906, size = 49, normalized size = 1.17 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arcoth}\left (\sqrt{x}\right ) + \frac{1}{6} \, x^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{x} - \frac{1}{4} \, \log \left (\sqrt{x} + 1\right ) + \frac{1}{4} \, \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 = 1.53167, size = 95, normalized size = 2.26 \begin{align*} \frac{1}{4} \,{\left (x^{2} - 1\right )} \log \left (\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\right ) + \frac{1}{6} \,{\left (x + 3\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 x \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 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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