Optimal. Leaf size=24 \[ \frac {x}{2-\log (2)+\frac {1}{4} x (3+x+\log (3 (2+x)))} \]
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Rubi [F] time = 0.79, 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 {64+32 x-12 x^2-4 x^3+(-32-16 x) \log (2)}{128+160 x+98 x^2+37 x^3+8 x^4+x^5+\left (-128-112 x-40 x^2-8 x^3\right ) \log (2)+(32+16 x) \log ^2(2)+\left (32 x+28 x^2+10 x^3+2 x^4+\left (-16 x-8 x^2\right ) \log (2)\right ) \log (6+3 x)+\left (2 x^2+x^3\right ) \log ^2(6+3 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 {4 \left (-3 x^2-x^3+8 (2-\log (2))+4 x (2-\log (2))\right )}{(2+x) \left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2} \, dx\\ &=4 \int \frac {-3 x^2-x^3+8 (2-\log (2))+4 x (2-\log (2))}{(2+x) \left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2} \, dx\\ &=4 \int \left (\frac {4}{(-2-x) \left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2}-\frac {x}{\left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2}-\frac {x^2}{\left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2}+\frac {2 (5-\log (4))}{\left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {x}{\left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2} \, dx\right )-4 \int \frac {x^2}{\left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2} \, dx+16 \int \frac {1}{(-2-x) \left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2} \, dx+(8 (5-\log (4))) \int \frac {1}{\left (3 x+x^2+8 \left (1-\frac {\log (2)}{2}\right )+x \log (3 (2+x))\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 2.22, size = 25, normalized size = 1.04 \begin {gather*} \frac {4 x}{8+3 x+x^2-4 \log (2)+x \log (3 (2+x))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 25, normalized size = 1.04 \begin {gather*} \frac {4 \, x}{x^{2} + x \log \left (3 \, x + 6\right ) + 3 \, x - 4 \, \log \relax (2) + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 25, normalized size = 1.04 \begin {gather*} \frac {4 \, x}{x^{2} + x \log \left (3 \, x + 6\right ) + 3 \, x - 4 \, \log \relax (2) + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 29, normalized size = 1.21
method | result | size |
norman | \(-\frac {4 x}{-\ln \left (6+3 x \right ) x -x^{2}+4 \ln \relax (2)-3 x -8}\) | \(29\) |
risch | \(-\frac {4 x}{-\ln \left (6+3 x \right ) x -x^{2}+4 \ln \relax (2)-3 x -8}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 26, normalized size = 1.08 \begin {gather*} \frac {4 \, x}{x^{2} + x {\left (\log \relax (3) + 3\right )} + x \log \left (x + 2\right ) - 4 \, \log \relax (2) + 8} \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 \relax (2)\,\left (16\,x+32\right )-32\,x+12\,x^2+4\,x^3-64}{160\,x+\ln \left (3\,x+6\right )\,\left (32\,x-\ln \relax (2)\,\left (8\,x^2+16\,x\right )+28\,x^2+10\,x^3+2\,x^4\right )+{\ln \relax (2)}^2\,\left (16\,x+32\right )-\ln \relax (2)\,\left (8\,x^3+40\,x^2+112\,x+128\right )+{\ln \left (3\,x+6\right )}^2\,\left (x^3+2\,x^2\right )+98\,x^2+37\,x^3+8\,x^4+x^5+128} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 24, normalized size = 1.00 \begin {gather*} \frac {4 x}{x^{2} + x \log {\left (3 x + 6 \right )} + 3 x - 4 \log {\relax (2 )} + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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