Integrand size = 36, antiderivative size = 27 \[ \int \frac {e^2 \left (-6 x-30 x^2-2 x^3\right )+2 \log (x)-\log ^2(x)}{e^2 x^2} \, dx=-x^2+\frac {\log ^2(x)}{e^2 x}-6 \log \left (e^{5 x} x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 14, 2341, 2342} \[ \int \frac {e^2 \left (-6 x-30 x^2-2 x^3\right )+2 \log (x)-\log ^2(x)}{e^2 x^2} \, dx=-x^2-30 x+\frac {\log ^2(x)}{e^2 x}-6 \log (x) \]
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Rule 12
Rule 14
Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {e^2 \left (-6 x-30 x^2-2 x^3\right )+2 \log (x)-\log ^2(x)}{x^2} \, dx}{e^2} \\ & = \frac {\int \left (-\frac {2 e^2 \left (3+15 x+x^2\right )}{x}+\frac {2 \log (x)}{x^2}-\frac {\log ^2(x)}{x^2}\right ) \, dx}{e^2} \\ & = -\left (2 \int \frac {3+15 x+x^2}{x} \, dx\right )-\frac {\int \frac {\log ^2(x)}{x^2} \, dx}{e^2}+\frac {2 \int \frac {\log (x)}{x^2} \, dx}{e^2} \\ & = -\frac {2}{e^2 x}-\frac {2 \log (x)}{e^2 x}+\frac {\log ^2(x)}{e^2 x}-2 \int \left (15+\frac {3}{x}+x\right ) \, dx-\frac {2 \int \frac {\log (x)}{x^2} \, dx}{e^2} \\ & = -30 x-x^2-6 \log (x)+\frac {\log ^2(x)}{e^2 x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {e^2 \left (-6 x-30 x^2-2 x^3\right )+2 \log (x)-\log ^2(x)}{e^2 x^2} \, dx=-\frac {30 e^2 x+e^2 x^2+6 e^2 \log (x)-\frac {\log ^2(x)}{x}}{e^2} \]
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Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-x^{2}-30 x +\frac {\ln \left (x \right )^{2} {\mathrm e}^{-2}}{x}-6 \ln \left (x \right )\) | \(24\) |
norman | \(\frac {\ln \left (x \right )^{2} {\mathrm e}^{-2}-x^{3}-6 x \ln \left (x \right )-30 x^{2}}{x}\) | \(30\) |
default | \({\mathrm e}^{-2} \left (-x^{2} {\mathrm e}^{2}-30 \,{\mathrm e}^{2} x +\frac {\ln \left (x \right )^{2}}{x}-6 \,{\mathrm e}^{2} \ln \left (x \right )\right )\) | \(33\) |
parallelrisch | \(-\frac {{\mathrm e}^{-2} \left (x^{3} {\mathrm e}^{2}+30 x^{2} {\mathrm e}^{2}+6 x \,{\mathrm e}^{2} \ln \left (x \right )-\ln \left (x \right )^{2}\right )}{x}\) | \(37\) |
parts | \(-x^{2}-30 x -6 \ln \left (x \right )+2 \,{\mathrm e}^{-2} \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )-{\mathrm e}^{-2} \left (-\frac {\ln \left (x \right )^{2}}{x}-\frac {2 \ln \left (x \right )}{x}-\frac {2}{x}\right )\) | \(61\) |
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {e^2 \left (-6 x-30 x^2-2 x^3\right )+2 \log (x)-\log ^2(x)}{e^2 x^2} \, dx=-\frac {{\left (6 \, x e^{2} \log \left (x\right ) + {\left (x^{3} + 30 \, x^{2}\right )} e^{2} - \log \left (x\right )^{2}\right )} e^{\left (-2\right )}}{x} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {e^2 \left (-6 x-30 x^2-2 x^3\right )+2 \log (x)-\log ^2(x)}{e^2 x^2} \, dx=- x^{2} - 30 x - 6 \log {\left (x \right )} + \frac {\log {\left (x \right )}^{2}}{x e^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {e^2 \left (-6 x-30 x^2-2 x^3\right )+2 \log (x)-\log ^2(x)}{e^2 x^2} \, dx=-{\left (x^{2} e^{2} + 30 \, x e^{2} + 6 \, e^{2} \log \left (x\right ) - \frac {\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 2}{x} + \frac {2 \, \log \left (x\right )}{x} + \frac {2}{x}\right )} e^{\left (-2\right )} \]
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Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^2 \left (-6 x-30 x^2-2 x^3\right )+2 \log (x)-\log ^2(x)}{e^2 x^2} \, dx=-\frac {{\left (x^{3} e^{2} + 30 \, x^{2} e^{2} + 6 \, x e^{2} \log \left (x\right ) - \log \left (x\right )^{2}\right )} e^{\left (-2\right )}}{x} \]
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Time = 11.67 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {e^2 \left (-6 x-30 x^2-2 x^3\right )+2 \log (x)-\log ^2(x)}{e^2 x^2} \, dx=\frac {{\mathrm {e}}^{-2}\,{\ln \left (x\right )}^2}{x}-6\,\ln \left (x\right )-x^2-30\,x \]
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