Integrand size = 80, antiderivative size = 25 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=\frac {e^x}{\left (-\frac {e}{12}+e^x-x+x^2\right ) \log (2)} \]
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\[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=\int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {12 e^x \left (12-e-36 x+12 x^2\right )}{\left (e-12 e^x-12 (-1+x) x\right )^2 \log (2)} \, dx \\ & = \frac {12 \int \frac {e^x \left (12-e-36 x+12 x^2\right )}{\left (e-12 e^x-12 (-1+x) x\right )^2} \, dx}{\log (2)} \\ & = \frac {12 \int \left (-\frac {\left (1-\frac {12}{e}\right ) e^{1+x}}{\left (e-12 e^x+12 x-12 x^2\right )^2}-\frac {36 e^x x}{\left (-e+12 e^x-12 x+12 x^2\right )^2}+\frac {12 e^x x^2}{\left (-e+12 e^x-12 x+12 x^2\right )^2}\right ) \, dx}{\log (2)} \\ & = \frac {144 \int \frac {e^x x^2}{\left (-e+12 e^x-12 x+12 x^2\right )^2} \, dx}{\log (2)}-\frac {432 \int \frac {e^x x}{\left (-e+12 e^x-12 x+12 x^2\right )^2} \, dx}{\log (2)}+\frac {\left (12 \left (-1+\frac {12}{e}\right )\right ) \int \frac {e^{1+x}}{\left (e-12 e^x+12 x-12 x^2\right )^2} \, dx}{\log (2)} \\ \end{align*}
Time = 1.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\frac {e-12 (-1+x) x}{\left (e-12 e^x-12 (-1+x) x\right ) \log (2)} \]
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Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
norman | \(-\frac {12 \,{\mathrm e}^{x}}{\ln \left (2\right ) \left (-12 x^{2}+{\mathrm e}-12 \,{\mathrm e}^{x}+12 x \right )}\) | \(26\) |
risch | \(-\frac {-12 x^{2}+{\mathrm e}+12 x}{\ln \left (2\right ) \left (-12 x^{2}+{\mathrm e}-12 \,{\mathrm e}^{x}+12 x \right )}\) | \(35\) |
parallelrisch | \(-\frac {-144 x^{2}+12 \,{\mathrm e}+144 x}{12 \ln \left (2\right ) \left (-12 x^{2}+{\mathrm e}-12 \,{\mathrm e}^{x}+12 x \right )}\) | \(37\) |
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Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\frac {12 \, x^{2} - 12 \, x - e}{{\left (12 \, x^{2} - 12 \, x - e\right )} \log \left (2\right ) + 12 \, e^{x} \log \left (2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=\frac {- 12 x^{2} + 12 x + e}{12 x^{2} \log {\left (2 \right )} - 12 x \log {\left (2 \right )} + 12 e^{x} \log {\left (2 \right )} - e \log {\left (2 \right )}} \]
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Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\frac {12 \, x^{2} - 12 \, x - e}{12 \, x^{2} \log \left (2\right ) - 12 \, x \log \left (2\right ) - e \log \left (2\right ) + 12 \, e^{x} \log \left (2\right )} \]
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\frac {12 \, x^{2} - 12 \, x - e}{12 \, x^{2} \log \left (2\right ) - 12 \, x \log \left (2\right ) - e \log \left (2\right ) + 12 \, e^{x} \log \left (2\right )} \]
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Timed out. \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\int \frac {{\mathrm {e}}^x\,\left (-144\,x^2+432\,x+12\,\mathrm {e}-144\right )}{144\,{\mathrm {e}}^{2\,x}\,\ln \left (2\right )+\ln \left (2\right )\,\left ({\mathrm {e}}^2+\mathrm {e}\,\left (24\,x-24\,x^2\right )+144\,x^2-288\,x^3+144\,x^4\right )-{\mathrm {e}}^x\,\ln \left (2\right )\,\left (-288\,x^2+288\,x+24\,\mathrm {e}\right )} \,d x \]
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