Optimal. Leaf size=20 \[ 2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (x+1) \]
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Rubi [A] time = 0.0075875, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5033, 31} \[ 2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (x+1) \]
Antiderivative was successfully verified.
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Rule 5033
Rule 31
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\int \frac{1}{1+x} \, dx\\ &=2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0068592, size = 20, normalized size = 1. \[ 2 \sqrt{x} \tan ^{-1}\left (\sqrt{x}\right )-\log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 17, normalized size = 0.9 \begin{align*} -\ln \left ( 1+x \right ) +2\,\arctan \left ( \sqrt{x} \right ) \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.926823, size = 22, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0133, size = 54, normalized size = 2.7 \begin{align*} 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.480636, size = 17, normalized size = 0.85 \begin{align*} 2 \sqrt{x} \operatorname{atan}{\left (\sqrt{x} \right )} - \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04905, size = 22, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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