Optimal. Leaf size=26 \[ -\frac {1}{2} \tan ^{-1}\left (\frac {1+x}{2}\right )+\frac {1}{2} \log \left (5+2 x+x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {648, 632, 210,
642} \begin {gather*} \frac {1}{2} \log \left (x^2+2 x+5\right )-\frac {1}{2} \text {ArcTan}\left (\frac {x+1}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x}{5+2 x+x^2} \, dx &=\frac {1}{2} \int \frac {2+2 x}{5+2 x+x^2} \, dx-\int \frac {1}{5+2 x+x^2} \, dx\\ &=\frac {1}{2} \log \left (5+2 x+x^2\right )+2 \text {Subst}\left (\int \frac {1}{-16-x^2} \, dx,x,2+2 x\right )\\ &=-\frac {1}{2} \tan ^{-1}\left (\frac {1+x}{2}\right )+\frac {1}{2} \log \left (5+2 x+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {1+x}{2}\right )+\frac {1}{2} \log \left (5+2 x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 21, normalized size = 0.81
method | result | size |
default | \(-\frac {\arctan \left (\frac {x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (x^{2}+2 x +5\right )}{2}\) | \(21\) |
risch | \(-\frac {\arctan \left (\frac {x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (x^{2}+2 x +5\right )}{2}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 20, normalized size = 0.77 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.98, size = 20, normalized size = 0.77 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 20, normalized size = 0.77 \begin {gather*} \frac {\log {\left (x^{2} + 2 x + 5 \right )}}{2} - \frac {\operatorname {atan}{\left (\frac {x}{2} + \frac {1}{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 20, normalized size = 0.77 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 20, normalized size = 0.77 \begin {gather*} \frac {\ln \left (x^2+2\,x+5\right )}{2}-\frac {\mathrm {atan}\left (\frac {x}{2}+\frac {1}{2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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