Optimal. Leaf size=28 \[ -e^3+\left (-1+e^2\right )^2-\frac {\log (x)}{3+2 x+\log ^2(x)} \]
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Rubi [F] time = 0.48, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-3-2 x+2 x \log (x)+\log ^2(x)}{9 x+12 x^2+4 x^3+\left (6 x+4 x^2\right ) \log ^2(x)+x \log ^4(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3-2 x+2 x \log (x)+\log ^2(x)}{x \left (3+2 x+\log ^2(x)\right )^2} \, dx\\ &=\int \left (\frac {2 (-3-2 x+x \log (x))}{x \left (3+2 x+\log ^2(x)\right )^2}+\frac {1}{x \left (3+2 x+\log ^2(x)\right )}\right ) \, dx\\ &=2 \int \frac {-3-2 x+x \log (x)}{x \left (3+2 x+\log ^2(x)\right )^2} \, dx+\int \frac {1}{x \left (3+2 x+\log ^2(x)\right )} \, dx\\ &=2 \int \left (-\frac {2}{\left (3+2 x+\log ^2(x)\right )^2}-\frac {3}{x \left (3+2 x+\log ^2(x)\right )^2}+\frac {\log (x)}{\left (3+2 x+\log ^2(x)\right )^2}\right ) \, dx+\int \frac {1}{x \left (3+2 x+\log ^2(x)\right )} \, dx\\ &=2 \int \frac {\log (x)}{\left (3+2 x+\log ^2(x)\right )^2} \, dx-4 \int \frac {1}{\left (3+2 x+\log ^2(x)\right )^2} \, dx-6 \int \frac {1}{x \left (3+2 x+\log ^2(x)\right )^2} \, dx+\int \frac {1}{x \left (3+2 x+\log ^2(x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\log (x)}{3+2 x+\log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\log \relax (x)}{\log \relax (x)^{2} + 2 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\log \relax (x)}{\log \relax (x)^{2} + 2 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 16, normalized size = 0.57
method | result | size |
norman | \(-\frac {\ln \relax (x )}{3+\ln \relax (x )^{2}+2 x}\) | \(16\) |
risch | \(-\frac {\ln \relax (x )}{3+\ln \relax (x )^{2}+2 x}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\log \relax (x)}{\log \relax (x)^{2} + 2 \, x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.26, size = 15, normalized size = 0.54 \begin {gather*} -\frac {\ln \relax (x)}{{\ln \relax (x)}^2+2\,x+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 14, normalized size = 0.50 \begin {gather*} - \frac {\log {\relax (x )}}{2 x + \log {\relax (x )}^{2} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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