Integrand size = 53, antiderivative size = 22 \[ \int \frac {e^{2 x^2+4 e^{4 x} x^2} \left (e^5 \left (-18+36 x^2\right )+e^{5+4 x} \left (72 x^2+144 x^3\right )\right )}{x^3} \, dx=\frac {9 e^{5+2 x \left (x+2 e^{4 x} x\right )}}{x^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(22)=44\).
Time = 0.41 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.23, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6873, 12, 2326} \[ \int \frac {e^{2 x^2+4 e^{4 x} x^2} \left (e^5 \left (-18+36 x^2\right )+e^{5+4 x} \left (72 x^2+144 x^3\right )\right )}{x^3} \, dx=\frac {9 e^{2 \left (2 e^{4 x}+1\right ) x^2+5} \left (4 e^{4 x} x^3+2 e^{4 x} x^2+x^2\right )}{x^3 \left (4 e^{4 x} x^2+\left (2 e^{4 x}+1\right ) x\right )} \]
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Rule 12
Rule 2326
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {18 e^{5+2 \left (1+2 e^{4 x}\right ) x^2} \left (-1+2 x^2+4 e^{4 x} x^2+8 e^{4 x} x^3\right )}{x^3} \, dx \\ & = 18 \int \frac {e^{5+2 \left (1+2 e^{4 x}\right ) x^2} \left (-1+2 x^2+4 e^{4 x} x^2+8 e^{4 x} x^3\right )}{x^3} \, dx \\ & = \frac {9 e^{5+2 \left (1+2 e^{4 x}\right ) x^2} \left (x^2+2 e^{4 x} x^2+4 e^{4 x} x^3\right )}{x^3 \left (\left (1+2 e^{4 x}\right ) x+4 e^{4 x} x^2\right )} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x^2+4 e^{4 x} x^2} \left (e^5 \left (-18+36 x^2\right )+e^{5+4 x} \left (72 x^2+144 x^3\right )\right )}{x^3} \, dx=\frac {9 e^{5+\left (2+4 e^{4 x}\right ) x^2}}{x^2} \]
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Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {9 \,{\mathrm e}^{4 x^{2} {\mathrm e}^{4 x}+2 x^{2}+5}}{x^{2}}\) | \(23\) |
parallelrisch | \(\frac {9 \,{\mathrm e}^{5} {\mathrm e}^{2 x^{2} \left (2 \,{\mathrm e}^{4 x}+1\right )}}{x^{2}}\) | \(23\) |
norman | \(\frac {9 \,{\mathrm e}^{5} {\mathrm e}^{4 x^{2} {\mathrm e}^{4 x}+2 x^{2}}}{x^{2}}\) | \(24\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{2 x^2+4 e^{4 x} x^2} \left (e^5 \left (-18+36 x^2\right )+e^{5+4 x} \left (72 x^2+144 x^3\right )\right )}{x^3} \, dx=\frac {9 \, e^{\left (2 \, {\left (x^{2} e^{5} + 2 \, x^{2} e^{\left (4 \, x + 5\right )}\right )} e^{\left (-5\right )} + 5\right )}}{x^{2}} \]
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Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {e^{2 x^2+4 e^{4 x} x^2} \left (e^5 \left (-18+36 x^2\right )+e^{5+4 x} \left (72 x^2+144 x^3\right )\right )}{x^3} \, dx=\frac {9 e^{5} e^{4 x^{2} e^{4 x} + 2 x^{2}}}{x^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x^2+4 e^{4 x} x^2} \left (e^5 \left (-18+36 x^2\right )+e^{5+4 x} \left (72 x^2+144 x^3\right )\right )}{x^3} \, dx=\frac {9 \, e^{\left (4 \, x^{2} e^{\left (4 \, x\right )} + 2 \, x^{2} + 5\right )}}{x^{2}} \]
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\[ \int \frac {e^{2 x^2+4 e^{4 x} x^2} \left (e^5 \left (-18+36 x^2\right )+e^{5+4 x} \left (72 x^2+144 x^3\right )\right )}{x^3} \, dx=\int { \frac {18 \, {\left ({\left (2 \, x^{2} - 1\right )} e^{5} + 4 \, {\left (2 \, x^{3} + x^{2}\right )} e^{\left (4 \, x + 5\right )}\right )} e^{\left (4 \, x^{2} e^{\left (4 \, x\right )} + 2 \, x^{2}\right )}}{x^{3}} \,d x } \]
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Time = 14.75 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {e^{2 x^2+4 e^{4 x} x^2} \left (e^5 \left (-18+36 x^2\right )+e^{5+4 x} \left (72 x^2+144 x^3\right )\right )}{x^3} \, dx=\frac {9\,{\mathrm {e}}^5\,{\mathrm {e}}^{2\,x^2}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{4\,x}}}{x^2} \]
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