Optimal. Leaf size=6 \[ \tan ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 65, 209}
\begin {gather*} \text {ArcTan}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 209
Rubi steps
\begin {align*} \int \frac {1}{2 \sqrt {x} (1+x)} \, dx &=\frac {1}{2} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=\tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 6, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 5, normalized size = 0.83
method | result | size |
derivativedivides | \(\arctan \left (\sqrt {x}\right )\) | \(5\) |
default | \(\arctan \left (\sqrt {x}\right )\) | \(5\) |
meijerg | \(\arctan \left (\sqrt {x}\right )\) | \(5\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x}+x -1}{1+x}\right )}{2}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 4, normalized size = 0.67 \begin {gather*} \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 4, normalized size = 0.67 \begin {gather*} \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 5, normalized size = 0.83 \begin {gather*} \operatorname {atan}{\left (\sqrt {x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.87, size = 4, normalized size = 0.67 \begin {gather*} \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 4, normalized size = 0.67 \begin {gather*} \mathrm {atan}\left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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