Integrand size = 16, antiderivative size = 19 \[ \int \frac {e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx=-\frac {e^{x^2}}{x}+2 e^{x^2} x \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6874, 2245, 2235, 2243} \[ \int \frac {e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx=2 e^{x^2} x-\frac {e^{x^2}}{x} \]
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Rule 2235
Rule 2243
Rule 2245
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^{x^2}}{x^2}+4 e^{x^2} x^2\right ) \, dx \\ & = 4 \int e^{x^2} x^2 \, dx+\int \frac {e^{x^2}}{x^2} \, dx \\ & = -\frac {e^{x^2}}{x}+2 e^{x^2} x \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx=\frac {e^{x^2} \left (-1+2 x^2\right )}{x} \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {{\mathrm e}^{x^{2}} \left (2 x^{2}-1\right )}{x}\) | \(16\) |
risch | \(\frac {{\mathrm e}^{x^{2}} \left (2 x^{2}-1\right )}{x}\) | \(16\) |
default | \(-\frac {{\mathrm e}^{x^{2}}}{x}+2 \,{\mathrm e}^{x^{2}} x\) | \(18\) |
norman | \(\frac {2 \,{\mathrm e}^{x^{2}} x^{2}-{\mathrm e}^{x^{2}}}{x}\) | \(21\) |
parallelrisch | \(\frac {2 \,{\mathrm e}^{x^{2}} x^{2}-{\mathrm e}^{x^{2}}}{x}\) | \(21\) |
meijerg | \(2 i \left (-i x \,{\mathrm e}^{x^{2}}+\frac {i \operatorname {erfi}\left (x \right ) \sqrt {\pi }}{2}\right )+\frac {i \left (\frac {2 i {\mathrm e}^{x^{2}}}{x}-2 i \operatorname {erfi}\left (x \right ) \sqrt {\pi }\right )}{2}\) | \(44\) |
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx=\frac {{\left (2 \, x^{2} - 1\right )} e^{\left (x^{2}\right )}}{x} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx=\frac {\left (2 x^{2} - 1\right ) e^{x^{2}}}{x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx=2 \, x e^{\left (x^{2}\right )} + i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) - \frac {\sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -x^{2}\right )}{2 \, x} \]
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none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx=\frac {2 \, x^{2} e^{\left (x^{2}\right )} - e^{\left (x^{2}\right )}}{x} \]
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx=\frac {{\mathrm {e}}^{x^2}\,\left (2\,x^2-1\right )}{x} \]
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