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 = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 43
Rule 5034
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 1.13, size = 19, normalized size = 0.66 \[ \frac {2}{3} \, x^{\frac {3}{2}} \operatorname {arccot}\left (\sqrt {x}\right ) + \frac {1}{3} \, x - \frac {1}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 30, normalized size = 1.03 \[ \frac {2}{3} \, x^{\frac {3}{2}} \arctan \left (\frac {1}{\sqrt {x}}\right ) - \frac {1}{3} \, x {\left (\frac {1}{x} - 1\right )} - \frac {1}{3} \, \log \relax (x) - \frac {1}{3} \, \log \left (\frac {1}{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.69 \[ \frac {x}{3}+\frac {2 x^{\frac {3}{2}} \mathrm {arccot}\left (\sqrt {x}\right )}{3}-\frac {\ln \left (x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 19, normalized size = 0.66 \[ \frac {2}{3} \, x^{\frac {3}{2}} \operatorname {arccot}\left (\sqrt {x}\right ) + \frac {1}{3} \, x - \frac {1}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {x}\,\mathrm {acot}\left (\sqrt {x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.27, size = 24, normalized size = 0.83 \[ \frac {2 x^{\frac {3}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{3} + \frac {x}{3} - \frac {\log {\left (x + 1 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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