Optimal. Leaf size=63 \[ -\frac {5 x+7}{3 \left (x^2+x+1\right )}+\frac {1}{2} \log \left (x^2+x+1\right )-\frac {2}{x+1}-\log (x+1)-\frac {25 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.12, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1646, 1628, 634, 618, 204, 628} \[ -\frac {5 x+7}{3 \left (x^2+x+1\right )}+\frac {1}{2} \log \left (x^2+x+1\right )-\frac {2}{x+1}-\log (x+1)-\frac {25 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1628
Rule 1646
Rubi steps
\begin {align*} \int \frac {-2-3 x+x^2}{(1+x)^2 \left (1+x+x^2\right )^2} \, dx &=-\frac {7+5 x}{3 \left (1+x+x^2\right )}+\frac {1}{3} \int \frac {-8-19 x-5 x^2}{(1+x)^2 \left (1+x+x^2\right )} \, dx\\ &=-\frac {7+5 x}{3 \left (1+x+x^2\right )}+\frac {1}{3} \int \left (\frac {6}{(1+x)^2}-\frac {3}{1+x}+\frac {-11+3 x}{1+x+x^2}\right ) \, dx\\ &=-\frac {2}{1+x}-\frac {7+5 x}{3 \left (1+x+x^2\right )}-\log (1+x)+\frac {1}{3} \int \frac {-11+3 x}{1+x+x^2} \, dx\\ &=-\frac {2}{1+x}-\frac {7+5 x}{3 \left (1+x+x^2\right )}-\log (1+x)+\frac {1}{2} \int \frac {1+2 x}{1+x+x^2} \, dx-\frac {25}{6} \int \frac {1}{1+x+x^2} \, dx\\ &=-\frac {2}{1+x}-\frac {7+5 x}{3 \left (1+x+x^2\right )}-\log (1+x)+\frac {1}{2} \log \left (1+x+x^2\right )+\frac {25}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac {2}{1+x}-\frac {7+5 x}{3 \left (1+x+x^2\right )}-\frac {25 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\log (1+x)+\frac {1}{2} \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 1.00 \[ -\frac {5 x+7}{3 \left (x^2+x+1\right )}+\frac {1}{2} \log \left (x^2+x+1\right )-\frac {2}{x+1}-\log (x+1)-\frac {25 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 69, normalized size = 1.10 \[ \frac {-11 x^2-18 x-13}{3 (x+1) \left (x^2+x+1\right )}+\frac {1}{2} \log \left (x^2+x+1\right )-\log (x+1)-\frac {25 \tan ^{-1}\left (\frac {2 x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 97, normalized size = 1.54 \[ -\frac {50 \, \sqrt {3} {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + 66 \, x^{2} - 9 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} \log \left (x^{2} + x + 1\right ) + 18 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + 108 \, x + 78}{18 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 72, normalized size = 1.14 \[ -\frac {25}{9} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (\frac {2}{x + 1} - 1\right )}\right ) + \frac {\frac {7}{x + 1} - 2}{3 \, {\left (\frac {1}{x + 1} - \frac {1}{{\left (x + 1\right )}^{2}} - 1\right )}} - \frac {2}{x + 1} + \frac {1}{2} \, \log \left (-\frac {1}{x + 1} + \frac {1}{{\left (x + 1\right )}^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 54, normalized size = 0.86
method | result | size |
default | \(\frac {-\frac {5 x}{3}-\frac {7}{3}}{x^{2}+x +1}+\frac {\ln \left (x^{2}+x +1\right )}{2}-\frac {25 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{9}-\frac {2}{1+x}-\ln \left (1+x \right )\) | \(54\) |
risch | \(\frac {-\frac {11}{3} x^{2}-6 x -\frac {13}{3}}{\left (x^{2}+x +1\right ) \left (1+x \right )}-\ln \left (1+x \right )+\frac {\ln \left (625 x^{2}+625 x +625\right )}{2}-\frac {25 \sqrt {3}\, \arctan \left (\frac {2 \left (25 x +\frac {25}{2}\right ) \sqrt {3}}{75}\right )}{9}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 59, normalized size = 0.94 \[ -\frac {25}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {11 \, x^{2} + 18 \, x + 13}{3 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}} + \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) - \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 73, normalized size = 1.16 \[ -\ln \left (x+1\right )-\frac {\frac {11\,x^2}{3}+6\,x+\frac {13}{3}}{x^3+2\,x^2+2\,x+1}+\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,25{}\mathrm {i}}{18}\right )-\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,25{}\mathrm {i}}{18}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 68, normalized size = 1.08 \[ \frac {- 11 x^{2} - 18 x - 13}{3 x^{3} + 6 x^{2} + 6 x + 3} - \log {\left (x + 1 \right )} + \frac {\log {\left (x^{2} + x + 1 \right )}}{2} - \frac {25 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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