Optimal. Leaf size=42 \[ \frac{x^{3/2}}{6}+\frac{1}{2} x^2 \cot ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x}}{2}+\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0085725, 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 = {5034, 50, 63, 203} \[ \frac{x^{3/2}}{6}+\frac{1}{2} x^2 \cot ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x}}{2}+\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5034
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int x \cot ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{2} x^2 \cot ^{-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 \cot ^{-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 \cot ^{-1}\left (\sqrt{x}\right )+\frac{1}{4} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=-\frac{\sqrt{x}}{2}+\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \cot ^{-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 \cot ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0106243, size = 33, normalized size = 0.79 \[ \frac{1}{6} \left (3 x^2 \cot ^{-1}\left (\sqrt{x}\right )+(x-3) \sqrt{x}+3 \tan ^{-1}\left (\sqrt{x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 27, normalized size = 0.6 \begin{align*}{\frac{1}{6}{x}^{{\frac{3}{2}}}}+{\frac{{x}^{2}}{2}{\rm arccot} \left (\sqrt{x}\right )}+{\frac{1}{2}\arctan \left ( \sqrt{x} \right ) }-{\frac{1}{2}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4242, size = 35, normalized size = 0.83 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arccot}\left (\sqrt{x}\right ) + \frac{1}{6} \, x^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{x} + \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28035, size = 72, normalized size = 1.71 \begin{align*} \frac{1}{2} \,{\left (x^{2} - 1\right )} \operatorname{arccot}\left (\sqrt{x}\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{acot}{\left (\sqrt{x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09093, size = 35, normalized size = 0.83 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\frac{1}{\sqrt{x}}\right ) + \frac{1}{6} \, x^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{x} + \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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