Optimal. Leaf size=19 \[ \left (64-x-\frac {x}{1+\log (3)}\right )^2+\log (x) \]
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Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.89, number of steps used = 5, number of rules used = 3, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6, 12, 14} \begin {gather*} \frac {x^2 (2+\log (3))^2}{(1+\log (3))^2}-\frac {128 x \left (2+\log ^2(3)+\log (27)\right )}{(1+\log (3))^2}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 14
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x \log ^2(3)+x (1+2 \log (3))} \, dx\\ &=\int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x \left (1+2 \log (3)+\log ^2(3)\right )} \, dx\\ &=\frac {\int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x} \, dx}{(1+\log (3))^2}\\ &=\frac {\int \left (2 x (2+\log (3))^2+\frac {1+2 \log (3)+\log ^2(3)}{x}-128 \left (2+\log ^2(3)+\log (27)\right )\right ) \, dx}{(1+\log (3))^2}\\ &=\frac {x^2 (2+\log (3))^2}{(1+\log (3))^2}-\frac {128 x \left (2+\log ^2(3)+\log (27)\right )}{(1+\log (3))^2}+\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.03, size = 40, normalized size = 2.11 \begin {gather*} \frac {x^2 (2+\log (3))^2-128 x \left (2+\log ^2(3)+\log (27)\right )+\left (1+\log ^2(3)+\log (9)\right ) \log (x)}{(1+\log (3))^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 58, normalized size = 3.05 \begin {gather*} \frac {{\left (x^{2} - 128 \, x\right )} \log \relax (3)^{2} + 4 \, x^{2} + 4 \, {\left (x^{2} - 96 \, x\right )} \log \relax (3) + {\left (\log \relax (3)^{2} + 2 \, \log \relax (3) + 1\right )} \log \relax (x) - 256 \, x}{\log \relax (3)^{2} + 2 \, \log \relax (3) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 97, normalized size = 5.11 \begin {gather*} \frac {x^{2} \log \relax (3)^{4} + 6 \, x^{2} \log \relax (3)^{3} - 128 \, x \log \relax (3)^{4} + 13 \, x^{2} \log \relax (3)^{2} - 640 \, x \log \relax (3)^{3} + 12 \, x^{2} \log \relax (3) - 1152 \, x \log \relax (3)^{2} + 4 \, x^{2} - 896 \, x \log \relax (3) - 256 \, x}{\log \relax (3)^{4} + 4 \, \log \relax (3)^{3} + 6 \, \log \relax (3)^{2} + 4 \, \log \relax (3) + 1} + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 40, normalized size = 2.11
method | result | size |
norman | \(\frac {\left (-128 \ln \relax (3)-256\right ) x +\frac {\left (\ln \relax (3)^{2}+4 \ln \relax (3)+4\right ) x^{2}}{\ln \relax (3)+1}}{\ln \relax (3)+1}+\ln \relax (x )\) | \(40\) |
default | \(\frac {x^{2} \ln \relax (3)^{2}-128 x \ln \relax (3)^{2}+4 x^{2} \ln \relax (3)-384 x \ln \relax (3)+4 x^{2}-256 x +\left (\ln \relax (3)^{2}+2 \ln \relax (3)+1\right ) \ln \relax (x )}{\ln \relax (3)^{2}+2 \ln \relax (3)+1}\) | \(63\) |
risch | \(\frac {\ln \relax (3)^{2} x^{2}}{\ln \relax (3)^{2}+2 \ln \relax (3)+1}-\frac {128 x \ln \relax (3)^{2}}{\ln \relax (3)^{2}+2 \ln \relax (3)+1}+\frac {4 \ln \relax (3) x^{2}}{\ln \relax (3)^{2}+2 \ln \relax (3)+1}-\frac {384 \ln \relax (3) x}{\ln \relax (3)^{2}+2 \ln \relax (3)+1}+\frac {4 x^{2}}{\ln \relax (3)^{2}+2 \ln \relax (3)+1}-\frac {256 x}{\ln \relax (3)^{2}+2 \ln \relax (3)+1}+\ln \relax (x )\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 44, normalized size = 2.32 \begin {gather*} \frac {{\left (\log \relax (3)^{2} + 4 \, \log \relax (3) + 4\right )} x^{2} - 128 \, {\left (\log \relax (3)^{2} + 3 \, \log \relax (3) + 2\right )} x}{\log \relax (3)^{2} + 2 \, \log \relax (3) + 1} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.04, size = 58, normalized size = 3.05 \begin {gather*} \ln \relax (x)-\frac {x\,\left (384\,\ln \relax (3)+128\,{\ln \relax (3)}^2+256\right )}{\ln \relax (9)+{\ln \relax (3)}^2+1}+\frac {x^2\,\left (8\,\ln \relax (3)+2\,{\ln \relax (3)}^2+8\right )}{2\,\left (\ln \relax (9)+{\ln \relax (3)}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 53, normalized size = 2.79 \begin {gather*} \frac {x^{2} \left (\log {\relax (3 )}^{2} + 4 + 4 \log {\relax (3 )}\right ) + x \left (- 384 \log {\relax (3 )} - 256 - 128 \log {\relax (3 )}^{2}\right ) + \left (1 + \log {\relax (3 )}\right )^{2} \log {\relax (x )}}{1 + \log {\relax (3 )}^{2} + 2 \log {\relax (3 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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