Integrand size = 54, antiderivative size = 19 \[ \int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x+2 x \log (3)+x \log ^2(3)} \, dx=\left (64-x-\frac {x}{1+\log (3)}\right )^2+\log (x) \]
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Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6, 12, 14} \[ \int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x+2 x \log (3)+x \log ^2(3)} \, dx=\frac {x^2 (2+\log (3))^2}{(1+\log (3))^2}-\frac {128 x \left (2+\log ^2(3)+\log (27)\right )}{(1+\log (3))^2}+\log (x) \]
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Rule 6
Rule 12
Rule 14
Rubi steps \begin{align*} \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{align*}
Leaf count is larger than twice the leaf count of optimal. \(40\) vs. \(2(19)=38\).
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x+2 x \log (3)+x \log ^2(3)} \, dx=\frac {x^2 (2+\log (3))^2-128 x \left (2+\log ^2(3)+\log (27)\right )+\left (1+\log ^2(3)+\log (9)\right ) \log (x)}{(1+\log (3))^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(19)=38\).
Time = 0.99 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11
method | result | size |
norman | \(\frac {\left (-128 \ln \left (3\right )-256\right ) x +\frac {\left (\ln \left (3\right )^{2}+4 \ln \left (3\right )+4\right ) x^{2}}{\ln \left (3\right )+1}}{\ln \left (3\right )+1}+\ln \left (x \right )\) | \(40\) |
default | \(\frac {x^{2} \ln \left (3\right )^{2}-128 x \ln \left (3\right )^{2}+4 x^{2} \ln \left (3\right )-384 x \ln \left (3\right )+4 x^{2}-256 x +\left (\ln \left (3\right )^{2}+2 \ln \left (3\right )+1\right ) \ln \left (x \right )}{\ln \left (3\right )^{2}+2 \ln \left (3\right )+1}\) | \(63\) |
parallelrisch | \(\frac {x^{2} \ln \left (3\right )^{2}+\ln \left (3\right )^{2} \ln \left (x \right )-128 x \ln \left (3\right )^{2}+4 x^{2} \ln \left (3\right )+2 \ln \left (3\right ) \ln \left (x \right )-384 x \ln \left (3\right )+4 x^{2}+\ln \left (x \right )-256 x}{\ln \left (3\right )^{2}+2 \ln \left (3\right )+1}\) | \(65\) |
risch | \(\frac {\ln \left (3\right )^{2} x^{2}}{\ln \left (3\right )^{2}+2 \ln \left (3\right )+1}-\frac {128 x \ln \left (3\right )^{2}}{\ln \left (3\right )^{2}+2 \ln \left (3\right )+1}+\frac {4 \ln \left (3\right ) x^{2}}{\ln \left (3\right )^{2}+2 \ln \left (3\right )+1}-\frac {384 \ln \left (3\right ) x}{\ln \left (3\right )^{2}+2 \ln \left (3\right )+1}+\frac {4 x^{2}}{\ln \left (3\right )^{2}+2 \ln \left (3\right )+1}-\frac {256 x}{\ln \left (3\right )^{2}+2 \ln \left (3\right )+1}+\ln \left (x \right )\) | \(111\) |
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (16) = 32\).
Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.05 \[ \int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x+2 x \log (3)+x \log ^2(3)} \, dx=\frac {{\left (x^{2} - 128 \, x\right )} \log \left (3\right )^{2} + 4 \, x^{2} + 4 \, {\left (x^{2} - 96 \, x\right )} \log \left (3\right ) + {\left (\log \left (3\right )^{2} + 2 \, \log \left (3\right ) + 1\right )} \log \left (x\right ) - 256 \, x}{\log \left (3\right )^{2} + 2 \, \log \left (3\right ) + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (14) = 28\).
Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.79 \[ \int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x+2 x \log (3)+x \log ^2(3)} \, dx=\frac {x^{2} \left (\log {\left (3 \right )}^{2} + 4 + 4 \log {\left (3 \right )}\right ) + x \left (- 384 \log {\left (3 \right )} - 256 - 128 \log {\left (3 \right )}^{2}\right ) + \left (1 + \log {\left (3 \right )}\right )^{2} \log {\left (x \right )}}{1 + \log {\left (3 \right )}^{2} + 2 \log {\left (3 \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (16) = 32\).
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.32 \[ \int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x+2 x \log (3)+x \log ^2(3)} \, dx=\frac {{\left (\log \left (3\right )^{2} + 4 \, \log \left (3\right ) + 4\right )} x^{2} - 128 \, {\left (\log \left (3\right )^{2} + 3 \, \log \left (3\right ) + 2\right )} x}{\log \left (3\right )^{2} + 2 \, \log \left (3\right ) + 1} + \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (16) = 32\).
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 5.11 \[ \int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x+2 x \log (3)+x \log ^2(3)} \, dx=\frac {x^{2} \log \left (3\right )^{4} + 6 \, x^{2} \log \left (3\right )^{3} - 128 \, x \log \left (3\right )^{4} + 13 \, x^{2} \log \left (3\right )^{2} - 640 \, x \log \left (3\right )^{3} + 12 \, x^{2} \log \left (3\right ) - 1152 \, x \log \left (3\right )^{2} + 4 \, x^{2} - 896 \, x \log \left (3\right ) - 256 \, x}{\log \left (3\right )^{4} + 4 \, \log \left (3\right )^{3} + 6 \, \log \left (3\right )^{2} + 4 \, \log \left (3\right ) + 1} + \log \left ({\left | x \right |}\right ) \]
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Time = 10.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.05 \[ \int \frac {1-256 x+8 x^2+\left (2-384 x+8 x^2\right ) \log (3)+\left (1-128 x+2 x^2\right ) \log ^2(3)}{x+2 x \log (3)+x \log ^2(3)} \, dx=\ln \left (x\right )-\frac {x\,\left (384\,\ln \left (3\right )+128\,{\ln \left (3\right )}^2+256\right )}{\ln \left (9\right )+{\ln \left (3\right )}^2+1}+\frac {x^2\,\left (8\,\ln \left (3\right )+2\,{\ln \left (3\right )}^2+8\right )}{2\,\left (\ln \left (9\right )+{\ln \left (3\right )}^2+1\right )} \]
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