Optimal. Leaf size=12 \[ \frac {2}{2+x}+\log (2+x) \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {27, 45}
\begin {gather*} \frac {2}{x+2}+\log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {x}{4+4 x+x^2} \, dx &=\int \frac {x}{(2+x)^2} \, dx\\ &=\int \left (-\frac {2}{(2+x)^2}+\frac {1}{2+x}\right ) \, dx\\ &=\frac {2}{2+x}+\log (2+x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {2}{2+x}+\log (2+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 13, normalized size = 1.08
method | result | size |
default | \(\frac {2}{2+x}+\ln \left (2+x \right )\) | \(13\) |
norman | \(\frac {2}{2+x}+\ln \left (2+x \right )\) | \(13\) |
risch | \(\frac {2}{2+x}+\ln \left (2+x \right )\) | \(13\) |
meijerg | \(-\frac {x}{2 \left (1+\frac {x}{2}\right )}+\ln \left (1+\frac {x}{2}\right )\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.84, size = 12, normalized size = 1.00 \begin {gather*} \frac {2}{x + 2} + \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 16, normalized size = 1.33 \begin {gather*} \frac {{\left (x + 2\right )} \log \left (x + 2\right ) + 2}{x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 8, normalized size = 0.67 \begin {gather*} \log {\left (x + 2 \right )} + \frac {2}{x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 13, normalized size = 1.08 \begin {gather*} \frac {2}{x + 2} + \log \left ({\left | x + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 12, normalized size = 1.00 \begin {gather*} \ln \left (x+2\right )+\frac {2}{x+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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