Integrand size = 105, antiderivative size = 27 \[ \int \frac {-120-120 e^2+e^{e^{3 x^2}} \left (-24-24 e^2-288 e^{3 x^2} x\right )}{125 e^{x+e^2 x}+75 e^{e^{3 x^2}+x+e^2 x}+15 e^{2 e^{3 x^2}+x+e^2 x}+e^{3 e^{3 x^2}+x+e^2 x}} \, dx=\frac {24 e^{-x-e^2 x}}{\left (5+e^{e^{3 x^2}}\right )^2} \]
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Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6820, 12, 2326} \[ \int \frac {-120-120 e^2+e^{e^{3 x^2}} \left (-24-24 e^2-288 e^{3 x^2} x\right )}{125 e^{x+e^2 x}+75 e^{e^{3 x^2}+x+e^2 x}+15 e^{2 e^{3 x^2}+x+e^2 x}+e^{3 e^{3 x^2}+x+e^2 x}} \, dx=\frac {24 e^{-\left (\left (1+e^2\right ) x\right )}}{\left (e^{e^{3 x^2}}+5\right )^2} \]
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Rule 12
Rule 2326
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {24 e^{-\left (\left (1+e^2\right ) x\right )} \left (-5 \left (1+e^2\right )-e^{e^{3 x^2}} \left (1+e^2\right )-12 e^{e^{3 x^2}+3 x^2} x\right )}{\left (5+e^{e^{3 x^2}}\right )^3} \, dx \\ & = 24 \int \frac {e^{-\left (\left (1+e^2\right ) x\right )} \left (-5 \left (1+e^2\right )-e^{e^{3 x^2}} \left (1+e^2\right )-12 e^{e^{3 x^2}+3 x^2} x\right )}{\left (5+e^{e^{3 x^2}}\right )^3} \, dx \\ & = \frac {24 e^{-\left (\left (1+e^2\right ) x\right )}}{\left (5+e^{e^{3 x^2}}\right )^2} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-120-120 e^2+e^{e^{3 x^2}} \left (-24-24 e^2-288 e^{3 x^2} x\right )}{125 e^{x+e^2 x}+75 e^{e^{3 x^2}+x+e^2 x}+15 e^{2 e^{3 x^2}+x+e^2 x}+e^{3 e^{3 x^2}+x+e^2 x}} \, dx=\frac {24 e^{-\left (\left (1+e^2\right ) x\right )}}{\left (5+e^{e^{3 x^2}}\right )^2} \]
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Time = 3.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {24 \,{\mathrm e}^{-\left ({\mathrm e}^{2}+1\right ) x}}{\left (5+{\mathrm e}^{{\mathrm e}^{3 x^{2}}}\right )^{2}}\) | \(22\) |
parallelrisch | \(\frac {24 \,{\mathrm e}^{-\left ({\mathrm e}^{2}+1\right ) x}}{{\mathrm e}^{2 \,{\mathrm e}^{3 x^{2}}}+10 \,{\mathrm e}^{{\mathrm e}^{3 x^{2}}}+25}\) | \(34\) |
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (23) = 46\).
Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {-120-120 e^2+e^{e^{3 x^2}} \left (-24-24 e^2-288 e^{3 x^2} x\right )}{125 e^{x+e^2 x}+75 e^{e^{3 x^2}+x+e^2 x}+15 e^{2 e^{3 x^2}+x+e^2 x}+e^{3 e^{3 x^2}+x+e^2 x}} \, dx=\frac {24 \, e^{\left (x e^{2} + x\right )}}{e^{\left (2 \, x e^{2} + 2 \, x + 2 \, e^{\left (3 \, x^{2}\right )}\right )} + 10 \, e^{\left (2 \, x e^{2} + 2 \, x + e^{\left (3 \, x^{2}\right )}\right )} + 25 \, e^{\left (2 \, x e^{2} + 2 \, x\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {-120-120 e^2+e^{e^{3 x^2}} \left (-24-24 e^2-288 e^{3 x^2} x\right )}{125 e^{x+e^2 x}+75 e^{e^{3 x^2}+x+e^2 x}+15 e^{2 e^{3 x^2}+x+e^2 x}+e^{3 e^{3 x^2}+x+e^2 x}} \, dx=\frac {24}{e^{x + x e^{2}} e^{2 e^{3 x^{2}}} + 10 e^{x + x e^{2}} e^{e^{3 x^{2}}} + 25 e^{x + x e^{2}}} \]
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none
Time = 0.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-120-120 e^2+e^{e^{3 x^2}} \left (-24-24 e^2-288 e^{3 x^2} x\right )}{125 e^{x+e^2 x}+75 e^{e^{3 x^2}+x+e^2 x}+15 e^{2 e^{3 x^2}+x+e^2 x}+e^{3 e^{3 x^2}+x+e^2 x}} \, dx=\frac {24}{e^{\left (x e^{2} + x + 2 \, e^{\left (3 \, x^{2}\right )}\right )} + 10 \, e^{\left (x e^{2} + x + e^{\left (3 \, x^{2}\right )}\right )} + 25 \, e^{\left (x e^{2} + x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.96 \[ \int \frac {-120-120 e^2+e^{e^{3 x^2}} \left (-24-24 e^2-288 e^{3 x^2} x\right )}{125 e^{x+e^2 x}+75 e^{e^{3 x^2}+x+e^2 x}+15 e^{2 e^{3 x^2}+x+e^2 x}+e^{3 e^{3 x^2}+x+e^2 x}} \, dx=\frac {24 \, {\left (x e^{\left (9 \, x^{2} + x e^{2} + x + e^{\left (3 \, x^{2}\right )}\right )} + 5 \, x e^{\left (9 \, x^{2} + x e^{2} + x\right )}\right )}}{x e^{\left (9 \, x^{2} + 2 \, x e^{2} + 2 \, x + 3 \, e^{\left (3 \, x^{2}\right )}\right )} + 15 \, x e^{\left (9 \, x^{2} + 2 \, x e^{2} + 2 \, x + 2 \, e^{\left (3 \, x^{2}\right )}\right )} + 75 \, x e^{\left (9 \, x^{2} + 2 \, x e^{2} + 2 \, x + e^{\left (3 \, x^{2}\right )}\right )} + 125 \, x e^{\left (9 \, x^{2} + 2 \, x e^{2} + 2 \, x\right )}} \]
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Time = 9.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-120-120 e^2+e^{e^{3 x^2}} \left (-24-24 e^2-288 e^{3 x^2} x\right )}{125 e^{x+e^2 x}+75 e^{e^{3 x^2}+x+e^2 x}+15 e^{2 e^{3 x^2}+x+e^2 x}+e^{3 e^{3 x^2}+x+e^2 x}} \, dx=\frac {24\,{\mathrm {e}}^{-x-x\,{\mathrm {e}}^2}}{10\,{\mathrm {e}}^{{\mathrm {e}}^{3\,x^2}}+{\mathrm {e}}^{2\,{\mathrm {e}}^{3\,x^2}}+25} \]
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