Integrand size = 21, antiderivative size = 17 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 e^{\frac {3}{2} \left (5-9 e^{2 x}\right )} \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2320, 2225} \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 e^{\frac {15}{2}-\frac {27 e^{2 x}}{2}} \]
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Rule 12
Rule 2225
Rule 2320
Rubi steps \begin{align*} \text {integral}& = -\left (81 \int e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx\right ) \\ & = -\left (\frac {81}{2} \text {Subst}\left (\int e^{\frac {15}{2}-\frac {27 x}{2}} \, dx,x,e^{2 x}\right )\right ) \\ & = 3 e^{\frac {15}{2}-\frac {27 e^{2 x}}{2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 e^{\frac {15}{2}-\frac {27 e^{2 x}}{2}} \]
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Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\) | \(12\) |
default | \(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\) | \(12\) |
norman | \(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\) | \(12\) |
risch | \(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\) | \(12\) |
parallelrisch | \(3 \,{\mathrm e}^{-\frac {27 \,{\mathrm e}^{2 x}}{2}+\frac {15}{2}}\) | \(12\) |
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Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 \, e^{\left (-\frac {27}{2} \, e^{\left (2 \, x\right )} + \frac {15}{2}\right )} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 e^{\frac {15}{2} - \frac {27 e^{2 x}}{2}} \]
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Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 \, e^{\left (-\frac {27}{2} \, e^{\left (2 \, x\right )} + \frac {15}{2}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3 \, e^{\left (-\frac {27}{2} \, e^{\left (2 \, x\right )} + \frac {15}{2}\right )} \]
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Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int -81 e^{\frac {1}{2} \left (15-27 e^{2 x}\right )+2 x} \, dx=3\,{\mathrm {e}}^{-\frac {27\,{\mathrm {e}}^{2\,x}}{2}}\,{\mathrm {e}}^{15/2} \]
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