Integrand size = 82, antiderivative size = 28 \[ \int \frac {e^{\frac {15-3 x^2+x \log \left (\frac {1}{16} \left (16+8 e^x x+e^{2 x} x^2\right )\right )}{3 x}} \left (-60-12 x^2+e^x \left (-15 x+2 x^2-x^3\right )\right )}{12 x^2+3 e^x x^3} \, dx=e^{\frac {5}{x}-x} \sqrt [3]{\left (1+\frac {e^x x}{4}\right )^2} \]
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Time = 0.44 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6838} \[ \int \frac {e^{\frac {15-3 x^2+x \log \left (\frac {1}{16} \left (16+8 e^x x+e^{2 x} x^2\right )\right )}{3 x}} \left (-60-12 x^2+e^x \left (-15 x+2 x^2-x^3\right )\right )}{12 x^2+3 e^x x^3} \, dx=\frac {e^{\frac {5-x^2}{x}} \sqrt [3]{e^{2 x} x^2+8 e^x x+16}}{2 \sqrt [3]{2}} \]
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Rule 6838
Rubi steps \begin{align*} \text {integral}& = \frac {e^{\frac {5-x^2}{x}} \sqrt [3]{16+8 e^x x+e^{2 x} x^2}}{2 \sqrt [3]{2}} \\ \end{align*}
Time = 2.46 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {15-3 x^2+x \log \left (\frac {1}{16} \left (16+8 e^x x+e^{2 x} x^2\right )\right )}{3 x}} \left (-60-12 x^2+e^x \left (-15 x+2 x^2-x^3\right )\right )}{12 x^2+3 e^x x^3} \, dx=\frac {e^{\frac {5}{x}-x} \sqrt [3]{\left (4+e^x x\right )^2}}{2 \sqrt [3]{2}} \]
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Time = 1.61 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {x \ln \left (\frac {{\mathrm e}^{2 x} x^{2}}{16}+\frac {{\mathrm e}^{x} x}{2}+1\right )-3 x^{2}+15}{3 x}}\) | \(33\) |
risch | \(\frac {\left ({\mathrm e}^{x} x +4\right )^{\frac {2}{3}} 2^{\frac {2}{3}} {\mathrm e}^{-\frac {i x \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x +4\right )^{2}\right )}^{3}-2 i x \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x +4\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{x} x +4\right )\right )+i x \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x +4\right )^{2}\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x +4\right )\right )}^{2}+6 x^{2}-30}{6 x}}}{4}\) | \(104\) |
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Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {15-3 x^2+x \log \left (\frac {1}{16} \left (16+8 e^x x+e^{2 x} x^2\right )\right )}{3 x}} \left (-60-12 x^2+e^x \left (-15 x+2 x^2-x^3\right )\right )}{12 x^2+3 e^x x^3} \, dx=e^{\left (-\frac {3 \, x^{2} - x \log \left (\frac {1}{16} \, x^{2} e^{\left (2 \, x\right )} + \frac {1}{2} \, x e^{x} + 1\right ) - 15}{3 \, x}\right )} \]
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Time = 34.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\frac {15-3 x^2+x \log \left (\frac {1}{16} \left (16+8 e^x x+e^{2 x} x^2\right )\right )}{3 x}} \left (-60-12 x^2+e^x \left (-15 x+2 x^2-x^3\right )\right )}{12 x^2+3 e^x x^3} \, dx=e^{\frac {- x^{2} + \frac {x \log {\left (\frac {x^{2} e^{2 x}}{16} + \frac {x e^{x}}{2} + 1 \right )}}{3} + 5}{x}} \]
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Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\frac {15-3 x^2+x \log \left (\frac {1}{16} \left (16+8 e^x x+e^{2 x} x^2\right )\right )}{3 x}} \left (-60-12 x^2+e^x \left (-15 x+2 x^2-x^3\right )\right )}{12 x^2+3 e^x x^3} \, dx=\frac {1}{4} \cdot 2^{\frac {2}{3}} {\left (x e^{x} + 4\right )}^{\frac {2}{3}} e^{\left (-x + \frac {5}{x}\right )} \]
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Time = 0.87 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {15-3 x^2+x \log \left (\frac {1}{16} \left (16+8 e^x x+e^{2 x} x^2\right )\right )}{3 x}} \left (-60-12 x^2+e^x \left (-15 x+2 x^2-x^3\right )\right )}{12 x^2+3 e^x x^3} \, dx=e^{\left (-x + \frac {5}{x} + \frac {1}{3} \, \log \left (\frac {1}{16} \, x^{2} e^{\left (2 \, x\right )} + \frac {1}{2} \, x e^{x} + 1\right )\right )} \]
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Time = 18.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{\frac {15-3 x^2+x \log \left (\frac {1}{16} \left (16+8 e^x x+e^{2 x} x^2\right )\right )}{3 x}} \left (-60-12 x^2+e^x \left (-15 x+2 x^2-x^3\right )\right )}{12 x^2+3 e^x x^3} \, dx=\frac {{16}^{2/3}\,{\mathrm {e}}^{\frac {5}{x}-x}\,{\left (x^2\,{\mathrm {e}}^{2\,x}+8\,x\,{\mathrm {e}}^x+16\right )}^{1/3}}{16} \]
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