Integrand size = 73, antiderivative size = 24 \[ \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{5 x \log ^4(2)-10 e^x x \log ^2(2) \log (3)+5 e^{2 x} x \log ^2(3)} \, dx=-\frac {3}{5 \left (e^{-x} \log ^2(2)-\log (3)\right )}+\log (x) \]
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Time = 0.60 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.110, Rules used = {6873, 12, 6874, 2320, 46, 36, 29, 31} \[ \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{5 x \log ^4(2)-10 e^x x \log ^2(2) \log (3)+5 e^{2 x} x \log ^2(3)} \, dx=\log (x)-\frac {3 \log ^2(2)}{5 \log (3) \left (\log ^2(2)-e^x \log (3)\right )} \]
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 46
Rule 2320
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{5 x \left (\log ^2(2)-e^x \log (3)\right )^2} \, dx \\ & = \frac {1}{5} \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{x \left (\log ^2(2)-e^x \log (3)\right )^2} \, dx \\ & = \frac {1}{5} \int \left (\frac {5}{x}-\frac {3 \log ^4(2)}{\log (3) \left (\log ^2(2)-e^x \log (3)\right )^2}+\frac {3 \log ^2(2)}{\log (3) \left (\log ^2(2)-e^x \log (3)\right )}\right ) \, dx \\ & = \log (x)+\frac {\left (3 \log ^2(2)\right ) \int \frac {1}{\log ^2(2)-e^x \log (3)} \, dx}{5 \log (3)}-\frac {\left (3 \log ^4(2)\right ) \int \frac {1}{\left (\log ^2(2)-e^x \log (3)\right )^2} \, dx}{5 \log (3)} \\ & = \log (x)+\frac {\left (3 \log ^2(2)\right ) \text {Subst}\left (\int \frac {1}{x \left (\log ^2(2)-x \log (3)\right )} \, dx,x,e^x\right )}{5 \log (3)}-\frac {\left (3 \log ^4(2)\right ) \text {Subst}\left (\int \frac {1}{x \left (\log ^2(2)-x \log (3)\right )^2} \, dx,x,e^x\right )}{5 \log (3)} \\ & = \log (x)+\frac {3}{5} \text {Subst}\left (\int \frac {1}{\log ^2(2)-x \log (3)} \, dx,x,e^x\right )+\frac {3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )}{5 \log (3)}-\frac {\left (3 \log ^4(2)\right ) \text {Subst}\left (\int \left (\frac {1}{x \log ^4(2)}+\frac {\log (3)}{\log ^2(2) \left (\log ^2(2)-x \log (3)\right )^2}+\frac {\log (3)}{\log ^4(2) \left (\log ^2(2)-x \log (3)\right )}\right ) \, dx,x,e^x\right )}{5 \log (3)} \\ & = -\frac {3 \log ^2(2)}{5 \log (3) \left (\log ^2(2)-e^x \log (3)\right )}+\log (x) \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{5 x \log ^4(2)-10 e^x x \log ^2(2) \log (3)+5 e^{2 x} x \log ^2(3)} \, dx=\frac {1}{5} \left (-\frac {3 \log ^2(2)}{\log (3) \left (\log ^2(2)-e^x \log (3)\right )}+5 \log (x)\right ) \]
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Time = 5.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
method | result | size |
norman | \(\frac {3 \,{\mathrm e}^{x}}{5 \left (\ln \left (3\right ) {\mathrm e}^{x}-\ln \left (2\right )^{2}\right )}+\ln \left (x \right )\) | \(22\) |
risch | \(\ln \left (x \right )+\frac {3 \ln \left (2\right )^{2}}{5 \ln \left (3\right ) \left (\ln \left (3\right ) {\mathrm e}^{x}-\ln \left (2\right )^{2}\right )}\) | \(28\) |
parallelrisch | \(\frac {5 \ln \left (x \right ) \ln \left (3\right )^{2} {\mathrm e}^{x}-5 \ln \left (x \right ) \ln \left (3\right ) \ln \left (2\right )^{2}+3 \ln \left (2\right )^{2}}{5 \ln \left (3\right ) \left (\ln \left (3\right ) {\mathrm e}^{x}-\ln \left (2\right )^{2}\right )}\) | \(48\) |
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{5 x \log ^4(2)-10 e^x x \log ^2(2) \log (3)+5 e^{2 x} x \log ^2(3)} \, dx=\frac {3 \, \log \left (2\right )^{2} + 5 \, {\left (e^{x} \log \left (3\right )^{2} - \log \left (3\right ) \log \left (2\right )^{2}\right )} \log \left (x\right )}{5 \, {\left (e^{x} \log \left (3\right )^{2} - \log \left (3\right ) \log \left (2\right )^{2}\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{5 x \log ^4(2)-10 e^x x \log ^2(2) \log (3)+5 e^{2 x} x \log ^2(3)} \, dx=\log {\left (x \right )} + \frac {3 \log {\left (2 \right )}^{2}}{5 e^{x} \log {\left (3 \right )}^{2} - 5 \log {\left (2 \right )}^{2} \log {\left (3 \right )}} \]
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none
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{5 x \log ^4(2)-10 e^x x \log ^2(2) \log (3)+5 e^{2 x} x \log ^2(3)} \, dx=\frac {3 \, \log \left (2\right )^{2}}{5 \, {\left (e^{x} \log \left (3\right )^{2} - \log \left (3\right ) \log \left (2\right )^{2}\right )}} + \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{5 x \log ^4(2)-10 e^x x \log ^2(2) \log (3)+5 e^{2 x} x \log ^2(3)} \, dx=\frac {5 \, e^{x} \log \left (3\right )^{2} \log \left (x\right ) - 5 \, \log \left (3\right ) \log \left (2\right )^{2} \log \left (x\right ) + 3 \, \log \left (2\right )^{2}}{5 \, {\left (e^{x} \log \left (3\right )^{2} - \log \left (3\right ) \log \left (2\right )^{2}\right )}} \]
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Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {5 \log ^4(2)+5 e^{2 x} \log ^2(3)+e^x \left (-3 x \log ^2(2)-10 \log ^2(2) \log (3)\right )}{5 x \log ^4(2)-10 e^x x \log ^2(2) \log (3)+5 e^{2 x} x \log ^2(3)} \, dx=\ln \left (x\right )+\frac {3\,{\mathrm {e}}^x}{5\,{\mathrm {e}}^x\,\ln \left (3\right )-5\,{\ln \left (2\right )}^2} \]
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