Optimal. Leaf size=16 \[ 2 \sqrt {x}-2 \arctan \left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {52, 65, 209}
\begin {gather*} 2 \sqrt {x}-2 \arctan \left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=2 \sqrt {x}-\int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=2 \sqrt {x}-2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {x}-2 \arctan \left (\sqrt {x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} 2 \sqrt {x}-2 \arctan \left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 13, normalized size = 0.81
method | result | size |
derivativedivides | \(2 \sqrt {x}-2 \arctan \left (\sqrt {x}\right )\) | \(13\) |
default | \(2 \sqrt {x}-2 \arctan \left (\sqrt {x}\right )\) | \(13\) |
meijerg | \(2 \sqrt {x}-2 \arctan \left (\sqrt {x}\right )\) | \(13\) |
risch | \(2 \sqrt {x}-2 \arctan \left (\sqrt {x}\right )\) | \(13\) |
trager | \(2 \sqrt {x}+\mathit {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \mathit {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x}-x +1}{x +1}\right )\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 12, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {x} - 2 \, \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 12, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {x} - 2 \, \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 14, normalized size = 0.88 \begin {gather*} 2 \sqrt {x} - 2 \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 = 0.43, size = 12, normalized size = 0.75 \begin {gather*} 2 \, \sqrt {x} - 2 \, \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 12, normalized size = 0.75 \begin {gather*} 2\,\sqrt {x}-2\,\mathrm {atan}\left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 22, normalized size = 1.38 \begin {gather*} \frac {\sqrt {x}\, \ln \left (\frac {x +1}{{| x +1|}}\right )}{4}+\frac {\sqrt {x}}{2} \end {gather*}
Warning: Unable to verify antiderivative.
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