Integrand size = 36, antiderivative size = 22 \[ \int \frac {2+x+(-4-3 x) \log (x \log (2))}{e^{20} \left (24 x^3+24 x^4+6 x^5\right )} \, dx=3+\frac {\log (x \log (2))}{6 e^{20} x^2 (2+x)} \]
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Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(22)=44\).
Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 1608, 27, 6874, 46, 2404, 2341, 2351, 31} \[ \int \frac {2+x+(-4-3 x) \log (x \log (2))}{e^{20} \left (24 x^3+24 x^4+6 x^5\right )} \, dx=\frac {\log (x \log (2))}{12 e^{20} x^2}+\frac {\log (x)}{48 e^{20}}-\frac {\log (x \log (2))}{24 e^{20} x}-\frac {x \log (x \log (2))}{48 e^{20} (x+2)} \]
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Rule 12
Rule 27
Rule 31
Rule 46
Rule 1608
Rule 2341
Rule 2351
Rule 2404
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {2+x+(-4-3 x) \log (x \log (2))}{24 x^3+24 x^4+6 x^5} \, dx}{e^{20}} \\ & = \frac {\int \frac {2+x+(-4-3 x) \log (x \log (2))}{x^3 \left (24+24 x+6 x^2\right )} \, dx}{e^{20}} \\ & = \frac {\int \frac {2+x+(-4-3 x) \log (x \log (2))}{6 x^3 (2+x)^2} \, dx}{e^{20}} \\ & = \frac {\int \frac {2+x+(-4-3 x) \log (x \log (2))}{x^3 (2+x)^2} \, dx}{6 e^{20}} \\ & = \frac {\int \left (\frac {1}{x^3 (2+x)}-\frac {(4+3 x) \log (x \log (2))}{x^3 (2+x)^2}\right ) \, dx}{6 e^{20}} \\ & = \frac {\int \frac {1}{x^3 (2+x)} \, dx}{6 e^{20}}-\frac {\int \frac {(4+3 x) \log (x \log (2))}{x^3 (2+x)^2} \, dx}{6 e^{20}} \\ & = \frac {\int \left (\frac {1}{2 x^3}-\frac {1}{4 x^2}+\frac {1}{8 x}-\frac {1}{8 (2+x)}\right ) \, dx}{6 e^{20}}-\frac {\int \left (\frac {\log (x \log (2))}{x^3}-\frac {\log (x \log (2))}{4 x^2}+\frac {\log (x \log (2))}{4 (2+x)^2}\right ) \, dx}{6 e^{20}} \\ & = -\frac {1}{24 e^{20} x^2}+\frac {1}{24 e^{20} x}+\frac {\log (x)}{48 e^{20}}-\frac {\log (2+x)}{48 e^{20}}+\frac {\int \frac {\log (x \log (2))}{x^2} \, dx}{24 e^{20}}-\frac {\int \frac {\log (x \log (2))}{(2+x)^2} \, dx}{24 e^{20}}-\frac {\int \frac {\log (x \log (2))}{x^3} \, dx}{6 e^{20}} \\ & = \frac {\log (x)}{48 e^{20}}-\frac {\log (2+x)}{48 e^{20}}+\frac {\log (x \log (2))}{12 e^{20} x^2}-\frac {\log (x \log (2))}{24 e^{20} x}-\frac {x \log (x \log (2))}{48 e^{20} (2+x)}+\frac {\int \frac {1}{2+x} \, dx}{48 e^{20}} \\ & = \frac {\log (x)}{48 e^{20}}+\frac {\log (x \log (2))}{12 e^{20} x^2}-\frac {\log (x \log (2))}{24 e^{20} x}-\frac {x \log (x \log (2))}{48 e^{20} (2+x)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {2+x+(-4-3 x) \log (x \log (2))}{e^{20} \left (24 x^3+24 x^4+6 x^5\right )} \, dx=\frac {\log (x \log (2))}{6 e^{20} x^2 (2+x)} \]
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Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {\ln \left (x \ln \left (2\right )\right ) {\mathrm e}^{-20}}{6 x^{2} \left (2+x \right )}\) | \(18\) |
norman | \(\frac {\ln \left (x \ln \left (2\right )\right ) {\mathrm e}^{-20}}{6 x^{2} \left (2+x \right )}\) | \(20\) |
parallelrisch | \(\frac {\ln \left (x \ln \left (2\right )\right ) {\mathrm e}^{-20}}{6 x^{2} \left (2+x \right )}\) | \(20\) |
derivativedivides | \(\frac {{\mathrm e}^{-20} \left (\frac {\ln \left (x \ln \left (2\right )\right ) \ln \left (2\right )}{8}-\frac {\ln \left (2\right ) \ln \left (x \ln \left (2\right )\right )}{4 x}-\frac {\ln \left (2\right )^{2} \ln \left (x \ln \left (2\right )\right ) x}{8 \left (2 \ln \left (2\right )+x \ln \left (2\right )\right )}+\frac {\ln \left (2\right ) \ln \left (x \ln \left (2\right )\right )}{2 x^{2}}\right )}{6 \ln \left (2\right )}\) | \(68\) |
default | \(\frac {{\mathrm e}^{-20} \left (\frac {\ln \left (x \ln \left (2\right )\right ) \ln \left (2\right )}{8}-\frac {\ln \left (2\right ) \ln \left (x \ln \left (2\right )\right )}{4 x}-\frac {\ln \left (2\right )^{2} \ln \left (x \ln \left (2\right )\right ) x}{8 \left (2 \ln \left (2\right )+x \ln \left (2\right )\right )}+\frac {\ln \left (2\right ) \ln \left (x \ln \left (2\right )\right )}{2 x^{2}}\right )}{6 \ln \left (2\right )}\) | \(68\) |
parts | \(\frac {{\mathrm e}^{-20} \left (-\frac {1}{4 x^{2}}+\frac {1}{4 x}+\frac {\ln \left (x \right )}{8}-\frac {\ln \left (2+x \right )}{8}\right )}{6}+\frac {{\mathrm e}^{-20} \ln \left (2 \ln \left (2\right )+x \ln \left (2\right )\right )}{48}-\frac {{\mathrm e}^{-20} \ln \left (2\right ) \ln \left (x \ln \left (2\right )\right ) x}{48 \left (2 \ln \left (2\right )+x \ln \left (2\right )\right )}+\frac {{\mathrm e}^{-20} \ln \left (x \ln \left (2\right )\right )}{12 x^{2}}+\frac {{\mathrm e}^{-20}}{24 x^{2}}-\frac {{\mathrm e}^{-20} \ln \left (x \ln \left (2\right )\right )}{24 x}-\frac {{\mathrm e}^{-20}}{24 x}\) | \(116\) |
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Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {2+x+(-4-3 x) \log (x \log (2))}{e^{20} \left (24 x^3+24 x^4+6 x^5\right )} \, dx=\frac {e^{\left (-20\right )} \log \left (x \log \left (2\right )\right )}{6 \, {\left (x^{3} + 2 \, x^{2}\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {2+x+(-4-3 x) \log (x \log (2))}{e^{20} \left (24 x^3+24 x^4+6 x^5\right )} \, dx=\frac {\log {\left (x \log {\left (2 \right )} \right )}}{6 x^{3} e^{20} + 12 x^{2} e^{20}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (19) = 38\).
Time = 0.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.86 \[ \int \frac {2+x+(-4-3 x) \log (x \log (2))}{e^{20} \left (24 x^3+24 x^4+6 x^5\right )} \, dx=\frac {1}{48} \, {\left (\frac {2 \, {\left (3 \, x^{2} + 3 \, x - 2\right )}}{x^{3} + 2 \, x^{2}} - \frac {2 \, x^{2} + {\left (x^{3} + 2 \, x^{2} - 8\right )} \log \left (x\right ) + 2 \, x - 8 \, \log \left (\log \left (2\right )\right ) - 4}{x^{3} + 2 \, x^{2}} - \frac {4 \, {\left (x + 1\right )}}{x^{2} + 2 \, x} + \log \left (x\right )\right )} e^{\left (-20\right )} \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {2+x+(-4-3 x) \log (x \log (2))}{e^{20} \left (24 x^3+24 x^4+6 x^5\right )} \, dx=\frac {1}{24} \, {\left (\frac {1}{x + 2} - \frac {x - 2}{x^{2}}\right )} e^{\left (-20\right )} \log \left (x \log \left (2\right )\right ) \]
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Time = 13.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {2+x+(-4-3 x) \log (x \log (2))}{e^{20} \left (24 x^3+24 x^4+6 x^5\right )} \, dx=\frac {\ln \left (\ln \left (2\right )\right )+\ln \left (x\right )}{6\,{\mathrm {e}}^{20}\,x^3+12\,{\mathrm {e}}^{20}\,x^2} \]
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