Optimal. Leaf size=28 \[ \frac {1}{\log \left (e^{-\frac {e^{-\frac {2}{3} e^{e^{20}} x+3 x^2}}{x}}\right )} \]
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Rubi [A]
time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.64, number of steps
used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 2326}
\begin {gather*} -\frac {e^{\frac {1}{3} \left (2 e^{e^{20}} x-9 x^2\right )} \left (e^{e^{20}} x-9 x^2\right )}{e^{e^{20}}-9 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int e^{\frac {1}{3} \left (2 e^{e^{20}} x-9 x^2\right )} \left (-3-2 e^{e^{20}} x+18 x^2\right ) \, dx\\ &=-\frac {e^{\frac {1}{3} \left (2 e^{e^{20}} x-9 x^2\right )} \left (e^{e^{20}} x-9 x^2\right )}{e^{e^{20}}-9 x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.43, size = 21, normalized size = 0.75 \begin {gather*} -e^{\frac {2 e^{e^{20}} x}{3}-3 x^2} x \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.24, size = 127, normalized size = 4.54
method | result | size |
risch | \(-x \,{\mathrm e}^{-\frac {x \left (-2 \,{\mathrm e}^{{\mathrm e}^{20}}+9 x \right )}{3}}\) | \(17\) |
gosper | \(-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}}\) | \(19\) |
norman | \(-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}}\) | \(19\) |
default | \(-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}}+\frac {2 \,{\mathrm e}^{{\mathrm e}^{20}} \left (-\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}}}{6}+\frac {{\mathrm e}^{{\mathrm e}^{20}} \sqrt {\pi }\, {\mathrm e}^{\frac {{\mathrm e}^{2 \,{\mathrm e}^{20}}}{27}} \sqrt {3}\, \erf \left (x \sqrt {3}-\frac {{\mathrm e}^{{\mathrm e}^{20}} \sqrt {3}}{9}\right )}{54}\right )}{3}+\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{{\mathrm e}^{20}}}{3}-3 x^{2}+{\mathrm e}^{20}}}{9}-\frac {{\mathrm e}^{{\mathrm e}^{20}} \sqrt {\pi }\, {\mathrm e}^{{\mathrm e}^{20}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{20}}}{27}} \sqrt {3}\, \erf \left (x \sqrt {3}-\frac {{\mathrm e}^{{\mathrm e}^{20}} \sqrt {3}}{9}\right )}{81}\) | \(127\) |
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 = 0.54, size = 267, normalized size = 9.54 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} x - \frac {1}{9} \, \sqrt {3} e^{\left (e^{20}\right )}\right ) e^{\left (\frac {1}{27} \, e^{\left (2 \, e^{20}\right )}\right )} - \frac {1}{81} i \, \sqrt {3} {\left (\frac {i \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (9 \, x - e^{\left (e^{20}\right )}\right )} {\left (\operatorname {erf}\left (\frac {1}{3} \, \sqrt {\frac {1}{3}} \sqrt {{\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}}\right ) - 1\right )} e^{\left (2 \, e^{20}\right )}}{\sqrt {{\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}}} - \frac {27 i \, \sqrt {3} \sqrt {\frac {1}{3}} {\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{27} \, {\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}\right )}{{\left ({\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}\right )}^{\frac {3}{2}}} - 6 i \, \sqrt {3} e^{\left (-\frac {1}{27} \, {\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2} + e^{20}\right )}\right )} e^{\left (\frac {1}{27} \, e^{\left (2 \, e^{20}\right )}\right )} + \frac {1}{81} i \, \sqrt {3} {\left (\frac {i \, \sqrt {3} \sqrt {\frac {1}{3}} \sqrt {\pi } {\left (9 \, x - e^{\left (e^{20}\right )}\right )} {\left (\operatorname {erf}\left (\frac {1}{3} \, \sqrt {\frac {1}{3}} \sqrt {{\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}}\right ) - 1\right )} e^{\left (e^{20}\right )}}{\sqrt {{\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}}} - 3 i \, \sqrt {3} e^{\left (-\frac {1}{27} \, {\left (9 \, x - e^{\left (e^{20}\right )}\right )}^{2}\right )}\right )} e^{\left (e^{20} + \frac {1}{27} \, e^{\left (2 \, e^{20}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 16, normalized size = 0.57 \begin {gather*} -x e^{\left (-3 \, x^{2} + \frac {2}{3} \, x e^{\left (e^{20}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 19, normalized size = 0.68 \begin {gather*} - x e^{- 3 x^{2} + \frac {2 x e^{e^{20}}}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 40, normalized size = 1.43 \begin {gather*} -\frac {1}{9} \, {\left (9 \, x + e^{\left (e^{20}\right )}\right )} e^{\left (-3 \, x^{2} + \frac {2}{3} \, x e^{\left (e^{20}\right )}\right )} + \frac {1}{9} \, e^{\left (-3 \, x^{2} + \frac {2}{3} \, x e^{\left (e^{20}\right )} + e^{20}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 16, normalized size = 0.57 \begin {gather*} -x\,{\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{20}}}{3}}\,{\mathrm {e}}^{-3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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