Optimal. Leaf size=18 \[ \log (x+1)+2 \sqrt {x} \cot ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 31
Rule 5034
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*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ \log (x+1)+2 \sqrt {x} \cot ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 14, normalized size = 0.78 \[ 2 \, \sqrt {x} \operatorname {arccot}\left (\sqrt {x}\right ) + \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 18, normalized size = 1.00 \[ 2 \, \sqrt {x} \arctan \left (\frac {1}{\sqrt {x}}\right ) + \log \relax (x) + \log \left (\frac {1}{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 15, normalized size = 0.83 \[ \ln \left (x +1\right )+2 \sqrt {x}\, \mathrm {arccot}\left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 14, normalized size = 0.78 \[ 2 \, \sqrt {x} \operatorname {arccot}\left (\sqrt {x}\right ) + \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.77, size = 14, normalized size = 0.78 \[ \ln \left (x+1\right )+2\,\sqrt {x}\,\mathrm {acot}\left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 17, normalized size = 0.94 \[ 2 \sqrt {x} \operatorname {acot}{\left (\sqrt {x} \right )} + \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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