Optimal. Leaf size=23 \[ -\frac {4}{5 (1+2 x)}+\log \left (\frac {1}{1+\frac {3}{x}+x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2074, 628} \begin {gather*} -\log \left (x^2+x+3\right )-\frac {4}{5 (2 x+1)}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{x}+\frac {8}{5 (1+2 x)^2}+\frac {-1-2 x}{3+x+x^2}\right ) \, dx\\ &=-\frac {4}{5 (1+2 x)}+\log (x)+\int \frac {-1-2 x}{3+x+x^2} \, dx\\ &=-\frac {4}{5 (1+2 x)}+\log (x)-\log \left (3+x+x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 1.17 \begin {gather*} \frac {1}{5} \left (-\frac {4}{1+2 x}+5 \log (x)-5 \log \left (3+x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 34, normalized size = 1.48 \begin {gather*} -\frac {5 \, {\left (2 \, x + 1\right )} \log \left (x^{2} + x + 3\right ) - 5 \, {\left (2 \, x + 1\right )} \log \relax (x) + 4}{5 \, {\left (2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 22, normalized size = 0.96 \begin {gather*} -\frac {4}{5 \, {\left (2 \, x + 1\right )}} - \log \left (x^{2} + x + 3\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.87
method | result | size |
risch | \(-\frac {2}{5 \left (\frac {1}{2}+x \right )}-\ln \left (x^{2}+x +3\right )+\ln \relax (x )\) | \(20\) |
default | \(-\frac {4}{5 \left (2 x +1\right )}+\ln \relax (x )-\ln \left (x^{2}+x +3\right )\) | \(22\) |
norman | \(\frac {8 x}{5 \left (2 x +1\right )}-\ln \left (x^{2}+x +3\right )+\ln \relax (x )\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 21, normalized size = 0.91 \begin {gather*} -\frac {4}{5 \, {\left (2 \, x + 1\right )}} - \log \left (x^{2} + x + 3\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 21, normalized size = 0.91 \begin {gather*} \ln \relax (x)-\ln \left (x^2+x+3\right )-\frac {2}{5\,\left (x+\frac {1}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 0.74 \begin {gather*} \log {\relax (x )} - \log {\left (x^{2} + x + 3 \right )} - \frac {4}{10 x + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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