Integrand size = 62, antiderivative size = 28 \[ \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{882 x^2+84 x^3+2 x^4} \, dx=1+\frac {e^3 \left (e^{e+\frac {x^2}{4}}+\frac {x}{21+x}\right )}{x} \]
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Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {1608, 27, 12, 6820, 2326} \[ \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{882 x^2+84 x^3+2 x^4} \, dx=\frac {e^{\frac {x^2}{4}+e+3}}{x}+\frac {e^3}{x+21} \]
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Rule 12
Rule 27
Rule 1608
Rule 2326
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{x^2 \left (882+84 x+2 x^2\right )} \, dx \\ & = \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{2 x^2 (21+x)^2} \, dx \\ & = \frac {1}{2} \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{x^2 (21+x)^2} \, dx \\ & = \frac {1}{2} \int \left (e^{3+e+\frac {x^2}{4}} \left (1-\frac {2}{x^2}\right )-\frac {2 e^3}{(21+x)^2}\right ) \, dx \\ & = \frac {e^3}{21+x}+\frac {1}{2} \int e^{3+e+\frac {x^2}{4}} \left (1-\frac {2}{x^2}\right ) \, dx \\ & = \frac {e^{3+e+\frac {x^2}{4}}}{x}+\frac {e^3}{21+x} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{882 x^2+84 x^3+2 x^4} \, dx=\frac {e^{3+e+\frac {x^2}{4}}}{x}+\frac {e^3}{21+x} \]
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Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {{\mathrm e}^{3}}{x +21}+\frac {{\mathrm e}^{3+{\mathrm e}+\frac {x^{2}}{4}}}{x}\) | \(24\) |
default | \(\frac {{\mathrm e}^{3+{\mathrm e}} {\mathrm e}^{\frac {x^{2}}{4}}}{x}+\frac {{\mathrm e}^{3}}{x +21}\) | \(25\) |
norman | \(\frac {x \,{\mathrm e}^{3}+x \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}+\frac {x^{2}}{4}}+21 \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}+\frac {x^{2}}{4}}}{x \left (x +21\right )}\) | \(41\) |
parallelrisch | \(\frac {2 x \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}+\frac {x^{2}}{4}}+2 x \,{\mathrm e}^{3}+42 \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}+\frac {x^{2}}{4}}}{2 x \left (x +21\right )}\) | \(44\) |
parts | \(\frac {{\mathrm e}^{3}}{x +21}+\frac {{\mathrm e}^{3} \left (-i {\mathrm e}^{{\mathrm e}} \sqrt {\pi }\, \operatorname {erf}\left (\frac {i x}{2}\right )-2 \,{\mathrm e}^{{\mathrm e}} \left (-\frac {{\mathrm e}^{\frac {x^{2}}{4}}}{x}-\frac {i \sqrt {\pi }\, \operatorname {erf}\left (\frac {i x}{2}\right )}{2}\right )\right )}{2}\) | \(57\) |
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{882 x^2+84 x^3+2 x^4} \, dx=\frac {x e^{3} + {\left (x + 21\right )} e^{\left (\frac {1}{4} \, x^{2} + e + 3\right )}}{x^{2} + 21 \, x} \]
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Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{882 x^2+84 x^3+2 x^4} \, dx=\frac {e^{3}}{x + 21} + \frac {e^{3} e^{\frac {x^{2}}{4} + e}}{x} \]
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{882 x^2+84 x^3+2 x^4} \, dx=\frac {e^{3}}{x + 21} + \frac {e^{\left (\frac {1}{4} \, x^{2} + e + 3\right )}}{x} \]
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{882 x^2+84 x^3+2 x^4} \, dx=\frac {x e^{3} + x e^{\left (\frac {1}{4} \, x^{2} + e + 3\right )} + 21 \, e^{\left (\frac {1}{4} \, x^{2} + e + 3\right )}}{x^{2} + 21 \, x} \]
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Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{882 x^2+84 x^3+2 x^4} \, dx=\frac {21\,{\mathrm {e}}^{\frac {x^2}{4}+\mathrm {e}+3}+x\,\left ({\mathrm {e}}^{\frac {x^2}{4}+\mathrm {e}+3}+{\mathrm {e}}^3\right )}{x\,\left (x+21\right )} \]
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