Integrand size = 19, antiderivative size = 15 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {\left (-5+e^x+\frac {x}{12}\right ) x}{e^4} \]
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Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(15)=30\).
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.20, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {12, 2207, 2225} \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {x^2}{12 e^4}-\frac {5 x}{e^4}-e^{x-4}+e^{x-4} (x+1) \]
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Rule 12
Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-30+x+e^x (6+6 x)\right ) \, dx}{6 e^4} \\ & = -\frac {5 x}{e^4}+\frac {x^2}{12 e^4}+\frac {\int e^x (6+6 x) \, dx}{6 e^4} \\ & = -\frac {5 x}{e^4}+\frac {x^2}{12 e^4}+e^{-4+x} (1+x)-\frac {\int e^x \, dx}{e^4} \\ & = -e^{-4+x}-\frac {5 x}{e^4}+\frac {x^2}{12 e^4}+e^{-4+x} (1+x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {-30 x+6 e^x x+\frac {x^2}{2}}{6 e^4} \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33
method | result | size |
risch | \(x \,{\mathrm e}^{x -4}-5 x \,{\mathrm e}^{-4}+\frac {x^{2} {\mathrm e}^{-4}}{12}\) | \(20\) |
default | \(\frac {{\mathrm e}^{-4} \left (-30 x +\frac {x^{2}}{2}+6 \,{\mathrm e}^{x} x \right )}{6}\) | \(21\) |
parallelrisch | \(\frac {{\mathrm e}^{-4} \left (-30 x +\frac {x^{2}}{2}+6 \,{\mathrm e}^{x} x \right )}{6}\) | \(21\) |
parts | \(x \,{\mathrm e}^{-4} {\mathrm e}^{x}+\frac {{\mathrm e}^{-4} \left (-30 x +\frac {1}{2} x^{2}\right )}{6}\) | \(25\) |
norman | \(x \,{\mathrm e}^{-4} {\mathrm e}^{x}-5 x \,{\mathrm e}^{-4}+\frac {x^{2} {\mathrm e}^{-4}}{12}\) | \(26\) |
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Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {1}{12} \, {\left (x^{2} + 12 \, x e^{x} - 60 \, x\right )} e^{\left (-4\right )} \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {x^{2}}{12 e^{4}} + \frac {x e^{x}}{e^{4}} - \frac {5 x}{e^{4}} \]
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Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {1}{12} \, {\left (x^{2} + 12 \, x e^{x} - 60 \, x\right )} e^{\left (-4\right )} \]
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Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {1}{12} \, {\left (x^{2} + 12 \, x e^{x} - 60 \, x\right )} e^{\left (-4\right )} \]
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Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {-30+x+e^x (6+6 x)}{6 e^4} \, dx=\frac {x\,{\mathrm {e}}^{-4}\,\left (x+12\,{\mathrm {e}}^x-60\right )}{12} \]
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