Optimal. Leaf size=25 \[ -\log \left (x^2+2 x+5\right )+x-\frac {3}{2} \tan ^{-1}\left (\frac {x+1}{2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {703, 634, 618, 204, 628} \[ -\log \left (x^2+2 x+5\right )+x-\frac {3}{2} \tan ^{-1}\left (\frac {x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 703
Rubi steps
\begin {align*} \int \frac {x^2}{5+2 x+x^2} \, dx &=x+\int \frac {-5-2 x}{5+2 x+x^2} \, dx\\ &=x-3 \int \frac {1}{5+2 x+x^2} \, dx-\int \frac {2+2 x}{5+2 x+x^2} \, dx\\ &=x-\log \left (5+2 x+x^2\right )+6 \operatorname {Subst}\left (\int \frac {1}{-16-x^2} \, dx,x,2+2 x\right )\\ &=x-\frac {3}{2} \tan ^{-1}\left (\frac {1+x}{2}\right )-\log \left (5+2 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 1.00 \[ -\log \left (x^2+2 x+5\right )+x-\frac {3}{2} \tan ^{-1}\left (\frac {x+1}{2}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 27, normalized size = 1.08 \[ -\log \left (x^2+2 x+5\right )+x-\frac {3}{2} \tan ^{-1}\left (\frac {x}{2}+\frac {1}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 21, normalized size = 0.84 \[ x - \frac {3}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 21, normalized size = 0.84 \[ x - \frac {3}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 22, normalized size = 0.88
method | result | size |
default | \(x -\frac {3 \arctan \left (\frac {1}{2}+\frac {x}{2}\right )}{2}-\ln \left (x^{2}+2 x +5\right )\) | \(22\) |
risch | \(x -\frac {3 \arctan \left (\frac {1}{2}+\frac {x}{2}\right )}{2}-\ln \left (x^{2}+2 x +5\right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 21, normalized size = 0.84 \[ x - \frac {3}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 21, normalized size = 0.84 \[ x-\ln \left (x^2+2\,x+5\right )-\frac {3\,\mathrm {atan}\left (\frac {x}{2}+\frac {1}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 22, normalized size = 0.88 \[ x - \log {\left (x^{2} + 2 x + 5 \right )} - \frac {3 \operatorname {atan}{\left (\frac {x}{2} + \frac {1}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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