Optimal. Leaf size=51 \[ \frac{x^{5/2}}{15}+\frac{x^{3/2}}{9}+\frac{1}{3} x^3 \coth ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{3}-\frac{1}{3} \tanh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0147834, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6098, 50, 63, 206} \[ \frac{x^{5/2}}{15}+\frac{x^{3/2}}{9}+\frac{1}{3} x^3 \coth ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{3}-\frac{1}{3} \tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6098
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int x^2 \coth ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{3} x^3 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \frac{x^{5/2}}{1-x} \, dx\\ &=\frac{x^{5/2}}{15}+\frac{1}{3} x^3 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \frac{x^{3/2}}{1-x} \, dx\\ &=\frac{x^{3/2}}{9}+\frac{x^{5/2}}{15}+\frac{1}{3} x^3 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \frac{\sqrt{x}}{1-x} \, dx\\ &=\frac{\sqrt{x}}{3}+\frac{x^{3/2}}{9}+\frac{x^{5/2}}{15}+\frac{1}{3} x^3 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \frac{1}{(1-x) \sqrt{x}} \, dx\\ &=\frac{\sqrt{x}}{3}+\frac{x^{3/2}}{9}+\frac{x^{5/2}}{15}+\frac{1}{3} x^3 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{3}+\frac{x^{3/2}}{9}+\frac{x^{5/2}}{15}+\frac{1}{3} x^3 \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{3} \tanh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0175619, size = 59, normalized size = 1.16 \[ \frac{1}{90} \left (6 x^{5/2}+10 x^{3/2}+30 x^3 \coth ^{-1}\left (\sqrt{x}\right )+30 \sqrt{x}+15 \log \left (1-\sqrt{x}\right )-15 \log \left (\sqrt{x}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 42, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{3}{\rm arccoth} \left (\sqrt{x}\right )}+{\frac{1}{15}{x}^{{\frac{5}{2}}}}+{\frac{1}{9}{x}^{{\frac{3}{2}}}}+{\frac{1}{3}\sqrt{x}}+{\frac{1}{6}\ln \left ( -1+\sqrt{x} \right ) }-{\frac{1}{6}\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 = 0.961345, size = 55, normalized size = 1.08 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcoth}\left (\sqrt{x}\right ) + \frac{1}{15} \, x^{\frac{5}{2}} + \frac{1}{9} \, x^{\frac{3}{2}} + \frac{1}{3} \, \sqrt{x} - \frac{1}{6} \, \log \left (\sqrt{x} + 1\right ) + \frac{1}{6} \, \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 = 1.60583, size = 111, normalized size = 2.18 \begin{align*} \frac{1}{6} \,{\left (x^{3} - 1\right )} \log \left (\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\right ) + \frac{1}{45} \,{\left (3 \, x^{2} + 5 \, x + 15\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^{2} \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^{2} \operatorname{arcoth}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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