Optimal. Leaf size=29 \[ \frac{2}{3} x^{3/2} \cot ^{-1}\left (\sqrt{x}\right )+\frac{x}{3}-\frac{1}{3} \log (x+1) \]
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Rubi [A] time = 0.0105779, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5034, 43} \[ \frac{2}{3} x^{3/2} \cot ^{-1}\left (\sqrt{x}\right )+\frac{x}{3}-\frac{1}{3} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 5034
Rule 43
Rubi steps
\begin{align*} \int \sqrt{x} \cot ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{2}{3} x^{3/2} \cot ^{-1}\left (\sqrt{x}\right )+\frac{1}{3} \int \frac{x}{1+x} \, dx\\ &=\frac{2}{3} x^{3/2} \cot ^{-1}\left (\sqrt{x}\right )+\frac{1}{3} \int \left (1+\frac{1}{-1-x}\right ) \, dx\\ &=\frac{x}{3}+\frac{2}{3} x^{3/2} \cot ^{-1}\left (\sqrt{x}\right )-\frac{1}{3} \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0108016, size = 25, normalized size = 0.86 \[ \frac{1}{3} \left (2 x^{3/2} \cot ^{-1}\left (\sqrt{x}\right )+x-\log (x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 20, normalized size = 0.7 \begin{align*}{\frac{x}{3}}+{\frac{2}{3}{x}^{{\frac{3}{2}}}{\rm arccot} \left (\sqrt{x}\right )}-{\frac{\ln \left ( x+1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973025, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \operatorname{arccot}\left (\sqrt{x}\right ) + \frac{1}{3} \, x - \frac{1}{3} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18903, size = 73, normalized size = 2.52 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \operatorname{arccot}\left (\sqrt{x}\right ) + \frac{1}{3} \, x - \frac{1}{3} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.33226, size = 24, normalized size = 0.83 \begin{align*} \frac{2 x^{\frac{3}{2}} \operatorname{acot}{\left (\sqrt{x} \right )}}{3} + \frac{x}{3} - \frac{\log{\left (x + 1 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10654, size = 26, normalized size = 0.9 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \arctan \left (\frac{1}{\sqrt{x}}\right ) + \frac{1}{3} \, x - \frac{1}{3} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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