Integrand size = 52, antiderivative size = 21 \[ \int \frac {-4+e^{6 x} (5+30 x)+\left (4-5 e^{6 x}\right ) \log \left (\frac {5 e^6}{-4+5 e^{6 x}}\right )}{-4+5 e^{6 x}} \, dx=x-x \log \left (\frac {e^6}{-\frac {4}{5}+e^{6 x}}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.15 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.71, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6874, 2215, 2221, 2317, 2438, 2320, 2441, 2352} \[ \int \frac {-4+e^{6 x} (5+30 x)+\left (4-5 e^{6 x}\right ) \log \left (\frac {5 e^6}{-4+5 e^{6 x}}\right )}{-4+5 e^{6 x}} \, dx=\frac {1}{6} \operatorname {PolyLog}\left (2,\frac {5 e^{6 x}}{4}\right )+\frac {1}{6} \operatorname {PolyLog}\left (2,1-\frac {5 e^{6 x}}{4}\right )-5 x+x \log \left (1-\frac {5 e^{6 x}}{4}\right )-\frac {1}{6} \log \left (\frac {5 e^{6 x}}{4}\right ) \log \left (-\frac {5}{4-5 e^{6 x}}\right ) \]
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Rule 2215
Rule 2221
Rule 2317
Rule 2320
Rule 2352
Rule 2438
Rule 2441
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-5+6 x+\frac {24 x}{-4+5 e^{6 x}}-\log \left (\frac {5}{-4+5 e^{6 x}}\right )\right ) \, dx \\ & = -5 x+3 x^2+24 \int \frac {x}{-4+5 e^{6 x}} \, dx-\int \log \left (\frac {5}{-4+5 e^{6 x}}\right ) \, dx \\ & = -5 x-\frac {1}{6} \text {Subst}\left (\int \frac {\log \left (\frac {5}{-4+5 x}\right )}{x} \, dx,x,e^{6 x}\right )+30 \int \frac {e^{6 x} x}{-4+5 e^{6 x}} \, dx \\ & = -5 x-\frac {1}{6} \log \left (\frac {5 e^{6 x}}{4}\right ) \log \left (-\frac {5}{4-5 e^{6 x}}\right )+x \log \left (1-\frac {5 e^{6 x}}{4}\right )-\frac {5}{6} \text {Subst}\left (\int \frac {\log \left (\frac {5 x}{4}\right )}{-4+5 x} \, dx,x,e^{6 x}\right )-\int \log \left (1-\frac {5 e^{6 x}}{4}\right ) \, dx \\ & = -5 x-\frac {1}{6} \log \left (\frac {5 e^{6 x}}{4}\right ) \log \left (-\frac {5}{4-5 e^{6 x}}\right )+x \log \left (1-\frac {5 e^{6 x}}{4}\right )+\frac {1}{6} \operatorname {PolyLog}\left (2,1-\frac {5 e^{6 x}}{4}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {\log \left (1-\frac {5 x}{4}\right )}{x} \, dx,x,e^{6 x}\right ) \\ & = -5 x-\frac {1}{6} \log \left (\frac {5 e^{6 x}}{4}\right ) \log \left (-\frac {5}{4-5 e^{6 x}}\right )+x \log \left (1-\frac {5 e^{6 x}}{4}\right )+\frac {1}{6} \operatorname {PolyLog}\left (2,\frac {5 e^{6 x}}{4}\right )+\frac {1}{6} \operatorname {PolyLog}\left (2,1-\frac {5 e^{6 x}}{4}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.81 \[ \int \frac {-4+e^{6 x} (5+30 x)+\left (4-5 e^{6 x}\right ) \log \left (\frac {5 e^6}{-4+5 e^{6 x}}\right )}{-4+5 e^{6 x}} \, dx=\frac {1}{6} \left (-30 x+18 x^2+6 x \log \left (1-\frac {4 e^{-6 x}}{5}\right )-\log \left (\frac {5 e^{6 x}}{4}\right ) \log \left (\frac {5}{-4+5 e^{6 x}}\right )-\operatorname {PolyLog}\left (2,\frac {4 e^{-6 x}}{5}\right )+\operatorname {PolyLog}\left (2,1-\frac {5 e^{6 x}}{4}\right )\right ) \]
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Time = 0.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67
method | result | size |
risch | \(x \ln \left ({\mathrm e}^{6 x}-\frac {4}{5}\right )-5 x\) | \(14\) |
norman | \(x -x \ln \left (\frac {5 \,{\mathrm e}^{6}}{5 \,{\mathrm e}^{6 x}-4}\right )\) | \(23\) |
parallelrisch | \(x -x \ln \left (\frac {5 \,{\mathrm e}^{6}}{5 \,{\mathrm e}^{6 x}-4}\right )\) | \(23\) |
default | \(\frac {\ln \left ({\mathrm e}^{6 x}\right )}{6}+\frac {\left (6 x -\ln \left (\frac {5 \,{\mathrm e}^{6 x}}{4}\right )\right ) \ln \left (1-\frac {5 \,{\mathrm e}^{6 x}}{4}\right )}{6}-\frac {\left (\ln \left (\frac {5 \,{\mathrm e}^{6}}{5 \,{\mathrm e}^{6 x}-4}\right )+\ln \left (5 \,{\mathrm e}^{6 x}-4\right )\right ) \ln \left ({\mathrm e}^{6 x}\right )}{6}+\frac {\ln \left (5 \,{\mathrm e}^{6 x}-4\right ) \ln \left (\frac {5 \,{\mathrm e}^{6 x}}{4}\right )}{6}\) | \(85\) |
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-4+e^{6 x} (5+30 x)+\left (4-5 e^{6 x}\right ) \log \left (\frac {5 e^6}{-4+5 e^{6 x}}\right )}{-4+5 e^{6 x}} \, dx=-x \log \left (\frac {5 \, e^{6}}{5 \, e^{\left (6 \, x\right )} - 4}\right ) + x \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-4+e^{6 x} (5+30 x)+\left (4-5 e^{6 x}\right ) \log \left (\frac {5 e^6}{-4+5 e^{6 x}}\right )}{-4+5 e^{6 x}} \, dx=- x \log {\left (\frac {5 e^{6}}{5 e^{6 x} - 4} \right )} + x \]
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Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-4+e^{6 x} (5+30 x)+\left (4-5 e^{6 x}\right ) \log \left (\frac {5 e^6}{-4+5 e^{6 x}}\right )}{-4+5 e^{6 x}} \, dx=-x {\left (\log \left (5\right ) + 6\right )} + x \log \left (5 \, e^{\left (6 \, x\right )} - 4\right ) + x \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-4+e^{6 x} (5+30 x)+\left (4-5 e^{6 x}\right ) \log \left (\frac {5 e^6}{-4+5 e^{6 x}}\right )}{-4+5 e^{6 x}} \, dx=-x \log \left (\frac {5}{5 \, e^{\left (6 \, x\right )} - 4}\right ) - 5 \, x \]
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Time = 12.85 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {-4+e^{6 x} (5+30 x)+\left (4-5 e^{6 x}\right ) \log \left (\frac {5 e^6}{-4+5 e^{6 x}}\right )}{-4+5 e^{6 x}} \, dx=-x\,\left (\ln \left (5\right )+\ln \left (\frac {1}{5\,{\mathrm {e}}^{6\,x}-4}\right )+5\right ) \]
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