Integrand size = 36, antiderivative size = 29 \[ \int \frac {-13+e^{4-x^2} \left (2+4 x^2\right )+e^{3+x} \left (x^2+x^3\right )}{x^2} \, dx=\frac {3+2 \left (5-e^{4-x^2}\right )}{x}+e^{3+x} x \]
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Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {14, 2326, 2207, 2225} \[ \int \frac {-13+e^{4-x^2} \left (2+4 x^2\right )+e^{3+x} \left (x^2+x^3\right )}{x^2} \, dx=-\frac {2 e^{4-x^2}}{x}+e^{x+3} (x+1)-e^{x+3}+\frac {13}{x} \]
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Rule 14
Rule 2207
Rule 2225
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 e^{4-x^2} \left (1+2 x^2\right )}{x^2}+\frac {-13+e^{3+x} x^2+e^{3+x} x^3}{x^2}\right ) \, dx \\ & = 2 \int \frac {e^{4-x^2} \left (1+2 x^2\right )}{x^2} \, dx+\int \frac {-13+e^{3+x} x^2+e^{3+x} x^3}{x^2} \, dx \\ & = -\frac {2 e^{4-x^2}}{x}+\int \left (-\frac {13}{x^2}+e^{3+x} (1+x)\right ) \, dx \\ & = \frac {13}{x}-\frac {2 e^{4-x^2}}{x}+\int e^{3+x} (1+x) \, dx \\ & = \frac {13}{x}-\frac {2 e^{4-x^2}}{x}+e^{3+x} (1+x)-\int e^{3+x} \, dx \\ & = -e^{3+x}+\frac {13}{x}-\frac {2 e^{4-x^2}}{x}+e^{3+x} (1+x) \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-13+e^{4-x^2} \left (2+4 x^2\right )+e^{3+x} \left (x^2+x^3\right )}{x^2} \, dx=\frac {13-2 e^{4-x^2}+e^{3+x} x^2}{x} \]
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Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86
method | result | size |
norman | \(\frac {13+x^{2} {\mathrm e}^{3+x}-2 \,{\mathrm e}^{-x^{2}+4}}{x}\) | \(25\) |
parallelrisch | \(\frac {13+x^{2} {\mathrm e}^{3+x}-2 \,{\mathrm e}^{-x^{2}+4}}{x}\) | \(25\) |
risch | \(\frac {13}{x}+{\mathrm e}^{3+x} x -\frac {2 \,{\mathrm e}^{-\left (-2+x \right ) \left (2+x \right )}}{x}\) | \(27\) |
parts | \({\mathrm e}^{3+x} \left (3+x \right )-3 \,{\mathrm e}^{3+x}+\frac {13}{x}+2 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-x^{2}}}{x}-\sqrt {\pi }\, \operatorname {erf}\left (x \right )\right )+2 \,{\mathrm e}^{4} \sqrt {\pi }\, \operatorname {erf}\left (x \right )\) | \(53\) |
default | \({\mathrm e}^{x} {\mathrm e}^{3}+{\mathrm e}^{3} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+\frac {13}{x}+2 \,{\mathrm e}^{4} \sqrt {\pi }\, \operatorname {erf}\left (x \right )+2 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-x^{2}}}{x}-\sqrt {\pi }\, \operatorname {erf}\left (x \right )\right )\) | \(56\) |
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none
Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-13+e^{4-x^2} \left (2+4 x^2\right )+e^{3+x} \left (x^2+x^3\right )}{x^2} \, dx=\frac {x^{2} e^{\left (x + 3\right )} - 2 \, e^{\left (-x^{2} + 4\right )} + 13}{x} \]
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Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {-13+e^{4-x^2} \left (2+4 x^2\right )+e^{3+x} \left (x^2+x^3\right )}{x^2} \, dx=x e^{x + 3} - \frac {2 e^{4 - x^{2}}}{x} + \frac {13}{x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {-13+e^{4-x^2} \left (2+4 x^2\right )+e^{3+x} \left (x^2+x^3\right )}{x^2} \, dx=2 \, \sqrt {\pi } \operatorname {erf}\left (x\right ) e^{4} + {\left (x e^{3} - e^{3}\right )} e^{x} - \frac {\sqrt {x^{2}} e^{4} \Gamma \left (-\frac {1}{2}, x^{2}\right )}{x} + \frac {13}{x} + e^{\left (x + 3\right )} \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-13+e^{4-x^2} \left (2+4 x^2\right )+e^{3+x} \left (x^2+x^3\right )}{x^2} \, dx=\frac {x^{2} e^{\left (x + 3\right )} - 2 \, e^{\left (-x^{2} + 4\right )} + 13}{x} \]
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Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-13+e^{4-x^2} \left (2+4 x^2\right )+e^{3+x} \left (x^2+x^3\right )}{x^2} \, dx=x\,{\mathrm {e}}^{x+3}-\frac {2\,{\mathrm {e}}^{4-x^2}-13}{x} \]
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