Integrand size = 28, antiderivative size = 22 \[ \int \left (8 e^{2 x+\frac {1}{4} \left (-4 e^{2 x}-\log (5)\right )}+8 x\right ) \, dx=4 \left (-\frac {e^{-e^{2 x}}}{\sqrt [4]{5}}+x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2320, 12, 2225} \[ \int \left (8 e^{2 x+\frac {1}{4} \left (-4 e^{2 x}-\log (5)\right )}+8 x\right ) \, dx=4 x^2-\frac {4 e^{-e^{2 x}}}{\sqrt [4]{5}} \]
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Rule 12
Rule 2225
Rule 2320
Rubi steps \begin{align*} \text {integral}& = 4 x^2+8 \int e^{2 x+\frac {1}{4} \left (-4 e^{2 x}-\log (5)\right )} \, dx \\ & = 4 x^2+4 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt [4]{5}} \, dx,x,e^{2 x}\right ) \\ & = 4 x^2+\frac {4 \text {Subst}\left (\int e^{-x} \, dx,x,e^{2 x}\right )}{\sqrt [4]{5}} \\ & = -\frac {4 e^{-e^{2 x}}}{\sqrt [4]{5}}+4 x^2 \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \left (8 e^{2 x+\frac {1}{4} \left (-4 e^{2 x}-\log (5)\right )}+8 x\right ) \, dx=-\frac {4 e^{-e^{2 x}}}{\sqrt [4]{5}}+4 x^2 \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
risch | \(4 x^{2}-\frac {4 \,5^{\frac {3}{4}} {\mathrm e}^{-{\mathrm e}^{2 x}}}{5}\) | \(19\) |
default | \(4 x^{2}-4 \,{\mathrm e}^{-{\mathrm e}^{2 x}-\frac {\ln \left (5\right )}{4}}\) | \(21\) |
norman | \(4 x^{2}-4 \,{\mathrm e}^{-{\mathrm e}^{2 x}-\frac {\ln \left (5\right )}{4}}\) | \(21\) |
parallelrisch | \(4 x^{2}-4 \,{\mathrm e}^{-{\mathrm e}^{2 x}-\frac {\ln \left (5\right )}{4}}\) | \(21\) |
parts | \(4 x^{2}-4 \,{\mathrm e}^{-{\mathrm e}^{2 x}-\frac {\ln \left (5\right )}{4}}\) | \(21\) |
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Time = 0.58 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \left (8 e^{2 x+\frac {1}{4} \left (-4 e^{2 x}-\log (5)\right )}+8 x\right ) \, dx=4 \, {\left (x^{2} e^{\left (2 \, x\right )} - e^{\left (2 \, x - e^{\left (2 \, x\right )} - \frac {1}{4} \, \log \left (5\right )\right )}\right )} e^{\left (-2 \, x\right )} \]
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Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (8 e^{2 x+\frac {1}{4} \left (-4 e^{2 x}-\log (5)\right )}+8 x\right ) \, dx=4 x^{2} - \frac {4 \cdot 5^{\frac {3}{4}} e^{- e^{2 x}}}{5} \]
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Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \left (8 e^{2 x+\frac {1}{4} \left (-4 e^{2 x}-\log (5)\right )}+8 x\right ) \, dx=4 \, x^{2} - \frac {4}{5} \cdot 5^{\frac {3}{4}} e^{\left (-e^{\left (2 \, x\right )}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \left (8 e^{2 x+\frac {1}{4} \left (-4 e^{2 x}-\log (5)\right )}+8 x\right ) \, dx=4 \, x^{2} - \frac {4}{5} \cdot 5^{\frac {3}{4}} e^{\left (-e^{\left (2 \, x\right )}\right )} \]
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Time = 11.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \left (8 e^{2 x+\frac {1}{4} \left (-4 e^{2 x}-\log (5)\right )}+8 x\right ) \, dx=4\,x^2-\frac {4\,5^{3/4}\,{\mathrm {e}}^{-{\mathrm {e}}^{2\,x}}}{5} \]
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