Integrand size = 44, antiderivative size = 24 \[ \int \frac {-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)}{e^2 \log (4)} \, dx=x-e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)} \]
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Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {12, 1607, 2227, 2207, 2225} \[ \int \frac {-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)}{e^2 \log (4)} \, dx=-e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)}+x \]
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Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)\right ) \, dx}{e^2 \log (4)} \\ & = x-\frac {x^2}{\log (4)}+\frac {\int e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \, dx}{e^2} \\ & = x-\frac {x^2}{\log (4)}+\frac {\int e^{\frac {32 x}{e^2}} \left (-2 e^2-32 x\right ) x \, dx}{e^2} \\ & = x-\frac {x^2}{\log (4)}+\frac {\int \left (-2 e^{2+\frac {32 x}{e^2}} x-32 e^{\frac {32 x}{e^2}} x^2\right ) \, dx}{e^2} \\ & = x-\frac {x^2}{\log (4)}-\frac {2 \int e^{2+\frac {32 x}{e^2}} x \, dx}{e^2}-\frac {32 \int e^{\frac {32 x}{e^2}} x^2 \, dx}{e^2} \\ & = x-\frac {1}{16} e^{2+\frac {32 x}{e^2}} x-e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)}+\frac {1}{16} \int e^{2+\frac {32 x}{e^2}} \, dx+2 \int e^{\frac {32 x}{e^2}} x \, dx \\ & = \frac {1}{512} e^{4+\frac {32 x}{e^2}}+x-e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)}-\frac {1}{16} e^2 \int e^{\frac {32 x}{e^2}} \, dx \\ & = x-e^{\frac {32 x}{e^2}} x^2-\frac {x^2}{\log (4)} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)}{e^2 \log (4)} \, dx=x \left (1-e^{\frac {32 x}{e^2}} x-\frac {x}{\log (4)}\right ) \]
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Time = 0.59 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
risch | \(x -\frac {x^{2}}{2 \ln \left (2\right )}-{\mathrm e}^{32 x \,{\mathrm e}^{-2}} x^{2}\) | \(23\) |
derivativedivides | \(\frac {256 x \ln \left (2\right )-128 x^{2}-256 \ln \left (2\right ) {\mathrm e}^{32 x \,{\mathrm e}^{-2}} x^{2}}{256 \ln \left (2\right )}\) | \(35\) |
norman | \(\left (x \,{\mathrm e}-x^{2} {\mathrm e} \,{\mathrm e}^{32 x \,{\mathrm e}^{-2}}-\frac {{\mathrm e} x^{2}}{2 \ln \left (2\right )}\right ) {\mathrm e}^{-1}\) | \(39\) |
default | \(\frac {{\mathrm e}^{-2} \left (x \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2} \ln \left (2\right ) {\mathrm e}^{32 x \,{\mathrm e}^{-2}} x^{2}-\frac {x^{2} {\mathrm e}^{2}}{2}\right )}{\ln \left (2\right )}\) | \(49\) |
parallelrisch | \(\frac {{\mathrm e}^{-2} \left (-x^{2} {\mathrm e}^{2}-2 \,{\mathrm e}^{2} \ln \left (2\right ) {\mathrm e}^{32 x \,{\mathrm e}^{-2}} x^{2}+2 x \,{\mathrm e}^{2} \ln \left (2\right )\right )}{2 \ln \left (2\right )}\) | \(51\) |
parts | \(x -\frac {x^{2}}{2 \ln \left (2\right )}-\frac {{\mathrm e}^{2} \left ({\mathrm e}^{2} \left (8 x \,{\mathrm e}^{-2} {\mathrm e}^{32 x \,{\mathrm e}^{-2}}-\frac {{\mathrm e}^{32 x \,{\mathrm e}^{-2}}}{4}\right )+{\mathrm e}^{2} \left (128 \,{\mathrm e}^{32 x \,{\mathrm e}^{-2}} x^{2} {\mathrm e}^{-4}-8 x \,{\mathrm e}^{-2} {\mathrm e}^{32 x \,{\mathrm e}^{-2}}+\frac {{\mathrm e}^{32 x \,{\mathrm e}^{-2}}}{4}\right )\right )}{128}\) | \(108\) |
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Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)}{e^2 \log (4)} \, dx=-\frac {2 \, x^{2} e^{\left (32 \, x e^{\left (-2\right )}\right )} \log \left (2\right ) + x^{2} - 2 \, x \log \left (2\right )}{2 \, \log \left (2\right )} \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)}{e^2 \log (4)} \, dx=- x^{2} e^{\frac {32 x}{e^{2}}} - \frac {x^{2}}{2 \log {\left (2 \right )}} + x \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)}{e^2 \log (4)} \, dx=-\frac {{\left (2 \, x^{2} e^{\left (32 \, x e^{\left (-2\right )} + 2\right )} \log \left (2\right ) + x^{2} e^{2} - 2 \, x e^{2} \log \left (2\right )\right )} e^{\left (-2\right )}}{2 \, \log \left (2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.96 \[ \int \frac {-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)}{e^2 \log (4)} \, dx=-\frac {{\left (256 \, x^{2} e^{2} - 512 \, x e^{2} \log \left (2\right ) + {\left ({\left (32 \, x e^{2} - e^{4}\right )} e^{\left (2 \, {\left (16 \, x + e^{2}\right )} e^{\left (-2\right )}\right )} + {\left (512 \, x^{2} e^{2} - 32 \, x e^{4} + e^{6}\right )} e^{\left (32 \, x e^{\left (-2\right )}\right )}\right )} \log \left (2\right )\right )} e^{\left (-2\right )}}{512 \, \log \left (2\right )} \]
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Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-2 e^2 x+e^2 \log (4)+e^{\frac {32 x}{e^2}} \left (-2 e^2 x-32 x^2\right ) \log (4)}{e^2 \log (4)} \, dx=x-\frac {x^2}{2\,\ln \left (2\right )}-x^2\,{\mathrm {e}}^{32\,x\,{\mathrm {e}}^{-2}} \]
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