Optimal. Leaf size=18 \[ \log (x+1)+2 \sqrt{x} \cot ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0068054, antiderivative size = 18, 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 = {5034, 31} \[ \log (x+1)+2 \sqrt{x} \cot ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5034
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \sqrt{x} \cot ^{-1}\left (\sqrt{x}\right )+\int \frac{1}{1+x} \, dx\\ &=2 \sqrt{x} \cot ^{-1}\left (\sqrt{x}\right )+\log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0068556, size = 18, normalized size = 1. \[ \log (x+1)+2 \sqrt{x} \cot ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 15, normalized size = 0.8 \begin{align*} \ln \left ( x+1 \right ) +2\,\sqrt{x}{\rm arccot} \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955033, size = 19, normalized size = 1.06 \begin{align*} 2 \, \sqrt{x} \operatorname{arccot}\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 = 2.20801, size = 54, normalized size = 3. \begin{align*} 2 \, \sqrt{x} \operatorname{arccot}\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 = 0.449524, size = 17, normalized size = 0.94 \begin{align*} 2 \sqrt{x} \operatorname{acot}{\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 = 1.09288, size = 19, normalized size = 1.06 \begin{align*} 2 \, \sqrt{x} \arctan \left (\frac{1}{\sqrt{x}}\right ) + \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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