Optimal. Leaf size=23 \[ -x+\log \left (3+3 \left (x+\frac {x+\log (3+2 x)}{x}\right )\right ) \]
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Rubi [A] time = 0.56, antiderivative size = 22, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 4, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6741, 6742, 43, 6684} \begin {gather*} \log \left (x^2+2 x+\log (2 x+3)\right )-x-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 6684
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x-3 x^2-5 x^3-2 x^4+\left (-3-5 x-2 x^2\right ) \log (3+2 x)}{x (3+2 x) \left (2 x+x^2+\log (3+2 x)\right )} \, dx\\ &=\int \left (\frac {-1-x}{x}+\frac {2 \left (4+5 x+2 x^2\right )}{(3+2 x) \left (2 x+x^2+\log (3+2 x)\right )}\right ) \, dx\\ &=2 \int \frac {4+5 x+2 x^2}{(3+2 x) \left (2 x+x^2+\log (3+2 x)\right )} \, dx+\int \frac {-1-x}{x} \, dx\\ &=\log \left (2 x+x^2+\log (3+2 x)\right )+\int \left (-1-\frac {1}{x}\right ) \, dx\\ &=-x-\log (x)+\log \left (2 x+x^2+\log (3+2 x)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 22, normalized size = 0.96 \begin {gather*} -x-\log (x)+\log \left (2 x+x^2+\log (3+2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 22, normalized size = 0.96 \begin {gather*} -x + \log \left (x^{2} + 2 \, x + \log \left (2 \, x + 3\right )\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 26, normalized size = 1.13 \begin {gather*} -x + \log \left (-x^{2} - 2 \, x - \log \left (2 \, x + 3\right )\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 22, normalized size = 0.96
method | result | size |
risch | \(-x -\ln \relax (x )+\ln \left (\ln \left (2 x +3\right )+x \left (2+x \right )\right )\) | \(22\) |
norman | \(-x -\ln \relax (x )+\ln \left (x^{2}+\ln \left (2 x +3\right )+2 x \right )\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 22, normalized size = 0.96 \begin {gather*} -x + \log \left (x^{2} + 2 \, x + \log \left (2 \, x + 3\right )\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \left (2\,x+3\right )\,\left (2\,x^2+5\,x+3\right )-2\,x+3\,x^2+5\,x^3+2\,x^4}{\ln \left (2\,x+3\right )\,\left (2\,x^2+3\,x\right )+6\,x^2+7\,x^3+2\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 19, normalized size = 0.83 \begin {gather*} - x - \log {\relax (x )} + \log {\left (x^{2} + 2 x + \log {\left (2 x + 3 \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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