Optimal. Leaf size=19 \[ -\frac {e^{x^2}}{x}+2 e^{x^2} x \]
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Rubi [A]
time = 0.07, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6874, 2245,
2235, 2243} \begin {gather*} 2 e^{x^2} x-\frac {e^{x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2243
Rule 2245
Rule 6874
Rubi steps
\begin {align*} \int \frac {e^{x^2} \left (1+4 x^4\right )}{x^2} \, dx &=\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*}
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Mathematica [A]
time = 0.08, size = 15, normalized size = 0.79 \begin {gather*} e^{x^2} \left (-\frac {1}{x}+2 x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 18, normalized size = 0.95
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 | \(\frac {2 \,{\mathrm e}^{x^{2}} x^{2}-{\mathrm e}^{x^{2}}}{x}\) | \(21\) |
meijerg | \(2 i \left (-i x \,{\mathrm e}^{x^{2}}+\frac {i \erfi \left (x \right ) \sqrt {\pi }}{2}\right )+\frac {i \left (\frac {2 i {\mathrm e}^{x^{2}}}{x}-2 i \erfi \left (x \right ) \sqrt {\pi }\right )}{2}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.31, size = 36, normalized size = 1.89 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\left (2 \, x^{2} - 1\right )} e^{\left (x^{2}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 12, normalized size = 0.63 \begin {gather*} \frac {\left (2 x^{2} - 1\right ) e^{x^{2}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.52, size = 20, normalized size = 1.05 \begin {gather*} \frac {2 \, x^{2} e^{\left (x^{2}\right )} - e^{\left (x^{2}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\mathrm {e}}^{x^2}\,\left (2\,x^2-1\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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