Optimal. Leaf size=39 \[ \frac {1-x}{4 \left (x^2+2 x+3\right )}+\frac {3 \tan ^{-1}\left (\frac {x+1}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1660, 12, 618, 204} \[ \frac {1-x}{4 \left (x^2+2 x+3\right )}+\frac {3 \tan ^{-1}\left (\frac {x+1}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 1660
Rubi steps
\begin {align*} \int \frac {1+x+x^2}{\left (3+2 x+x^2\right )^2} \, dx &=\frac {1-x}{4 \left (3+2 x+x^2\right )}+\frac {1}{8} \int \frac {6}{3+2 x+x^2} \, dx\\ &=\frac {1-x}{4 \left (3+2 x+x^2\right )}+\frac {3}{4} \int \frac {1}{3+2 x+x^2} \, dx\\ &=\frac {1-x}{4 \left (3+2 x+x^2\right )}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,2+2 x\right )\\ &=\frac {1-x}{4 \left (3+2 x+x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {1+x}{\sqrt {2}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 1.00 \[ \frac {1-x}{4 \left (x^2+2 x+3\right )}+\frac {3 \tan ^{-1}\left (\frac {x+1}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 39, normalized size = 1.00 \[ \frac {3 \, \sqrt {2} {\left (x^{2} + 2 \, x + 3\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) - 2 \, x + 2}{8 \, {\left (x^{2} + 2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 30, normalized size = 0.77 \[ \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) - \frac {x - 1}{4 \, {\left (x^{2} + 2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 0.87 \[ \frac {3 \sqrt {2}\, \arctan \left (\frac {\left (2 x +2\right ) \sqrt {2}}{4}\right )}{8}+\frac {-\frac {x}{4}+\frac {1}{4}}{x^{2}+2 x +3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 30, normalized size = 0.77 \[ \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )}\right ) - \frac {x - 1}{4 \, {\left (x^{2} + 2 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.84, size = 36, normalized size = 0.92 \[ \frac {3\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}+\frac {\sqrt {2}}{2}\right )}{8}-\frac {\frac {x}{4}-\frac {1}{4}}{x^2+2\,x+3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 37, normalized size = 0.95 \[ \frac {1 - x}{4 x^{2} + 8 x + 12} + \frac {3 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} + \frac {\sqrt {2}}{2} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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