Optimal. Leaf size=20 \[ 2 \sqrt {x} \tan ^{-1}\left (\sqrt {x}\right )-\log (x+1) \]
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Rubi [A] time = 0.01, antiderivative size = 20, 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 = {5033, 31} \[ 2 \sqrt {x} \tan ^{-1}\left (\sqrt {x}\right )-\log (x+1) \]
Antiderivative was successfully verified.
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Rule 31
Rule 5033
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*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ 2 \sqrt {x} \tan ^{-1}\left (\sqrt {x}\right )-\log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 16, normalized size = 0.80 \[ 2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 16, normalized size = 0.80 \[ 2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 17, normalized size = 0.85 \[ -\ln \left (x +1\right )+2 \sqrt {x}\, \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 16, normalized size = 0.80 \[ 2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 16, normalized size = 0.80 \[ 2\,\sqrt {x}\,\mathrm {atan}\left (\sqrt {x}\right )-\ln \left (x+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 17, normalized size = 0.85 \[ 2 \sqrt {x} \operatorname {atan}{\left (\sqrt {x} \right )} - \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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