Optimal. Leaf size=29 \[ -\frac {x}{3}+\frac {2}{3} x^{3/2} \text {ArcTan}\left (\sqrt {x}\right )+\frac {1}{3} \log (1+x) \]
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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 = {4946, 45}
\begin {gather*} \frac {2}{3} x^{3/2} \text {ArcTan}\left (\sqrt {x}\right )-\frac {x}{3}+\frac {1}{3} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 4946
Rubi steps
\begin {align*} \int \sqrt {x} \tan ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {2}{3} x^{3/2} \tan ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x}{1+x} \, dx\\ &=\frac {2}{3} x^{3/2} \tan ^{-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} \tan ^{-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 \begin {gather*} \frac {1}{3} \left (-x+2 x^{3/2} \text {ArcTan}\left (\sqrt {x}\right )+\log (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 20, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {x}{3}+\frac {2 x^{\frac {3}{2}} \arctan \left (\sqrt {x}\right )}{3}+\frac {\ln \left (1+x \right )}{3}\) | \(20\) |
default | \(-\frac {x}{3}+\frac {2 x^{\frac {3}{2}} \arctan \left (\sqrt {x}\right )}{3}+\frac {\ln \left (1+x \right )}{3}\) | \(20\) |
meijerg | \(-\frac {x}{3}+\frac {2 x^{\frac {3}{2}} \arctan \left (\sqrt {x}\right )}{3}+\frac {\ln \left (1+x \right )}{3}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 19, normalized size = 0.66 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} \arctan \left (\sqrt {x}\right ) - \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.45, size = 19, normalized size = 0.66 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} \arctan \left (\sqrt {x}\right ) - \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.78, size = 24, normalized size = 0.83 \begin {gather*} \frac {2 x^{\frac {3}{2}} \operatorname {atan}{\left (\sqrt {x} \right )}}{3} - \frac {x}{3} + \frac {\log {\left (x + 1 \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 19, normalized size = 0.66 \begin {gather*} \frac {2}{3} \, x^{\frac {3}{2}} \arctan \left (\sqrt {x}\right ) - \frac {1}{3} \, x + \frac {1}{3} \, \log \left (x + 1\right ) \end {gather*}
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 \begin {gather*} \int \sqrt {x}\,\mathrm {atan}\left (\sqrt {x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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