Optimal. Leaf size=66 \[ \frac {147}{32} \sqrt {\pi } \text {erf}(x)-\frac {147}{16} e^{-x^2} x-\frac {1}{2} e^{-x^2} x^7-\frac {9}{4} e^{-x^2} x^5-\frac {49}{8} e^{-x^2} x^3 \]
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Rubi [A] time = 0.18, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1594, 2226, 2212, 2205} \[ \frac {147}{32} \sqrt {\pi } \text {Erf}(x)-\frac {1}{2} e^{-x^2} x^7-\frac {9}{4} e^{-x^2} x^5-\frac {49}{8} e^{-x^2} x^3-\frac {147}{16} e^{-x^2} x \]
Antiderivative was successfully verified.
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Rule 1594
Rule 2205
Rule 2212
Rule 2226
Rubi steps
\begin {align*} \int e^{-x^2} \left (x^4+x^6+x^8\right ) \, dx &=\int e^{-x^2} x^4 \left (1+x^2+x^4\right ) \, dx\\ &=\int \left (e^{-x^2} x^4+e^{-x^2} x^6+e^{-x^2} x^8\right ) \, dx\\ &=\int e^{-x^2} x^4 \, dx+\int e^{-x^2} x^6 \, dx+\int e^{-x^2} x^8 \, dx\\ &=-\frac {1}{2} e^{-x^2} x^3-\frac {1}{2} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {3}{2} \int e^{-x^2} x^2 \, dx+\frac {5}{2} \int e^{-x^2} x^4 \, dx+\frac {7}{2} \int e^{-x^2} x^6 \, dx\\ &=-\frac {3}{4} e^{-x^2} x-\frac {7}{4} e^{-x^2} x^3-\frac {9}{4} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {3}{4} \int e^{-x^2} \, dx+\frac {15}{4} \int e^{-x^2} x^2 \, dx+\frac {35}{4} \int e^{-x^2} x^4 \, dx\\ &=-\frac {21}{8} e^{-x^2} x-\frac {49}{8} e^{-x^2} x^3-\frac {9}{4} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {3}{8} \sqrt {\pi } \text {erf}(x)+\frac {15}{8} \int e^{-x^2} \, dx+\frac {105}{8} \int e^{-x^2} x^2 \, dx\\ &=-\frac {147}{16} e^{-x^2} x-\frac {49}{8} e^{-x^2} x^3-\frac {9}{4} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {21}{16} \sqrt {\pi } \text {erf}(x)+\frac {105}{16} \int e^{-x^2} \, dx\\ &=-\frac {147}{16} e^{-x^2} x-\frac {49}{8} e^{-x^2} x^3-\frac {9}{4} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {147}{32} \sqrt {\pi } \text {erf}(x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.62 \[ \frac {1}{32} \left (147 \sqrt {\pi } \text {erf}(x)-2 e^{-x^2} x \left (8 x^6+36 x^4+98 x^2+147\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 35, normalized size = 0.53 \[ -\frac {1}{16} \, {\left (8 \, x^{7} + 36 \, x^{5} + 98 \, x^{3} + 147 \, x\right )} e^{\left (-x^{2}\right )} + \frac {147}{32} \, \sqrt {\pi } \operatorname {erf}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 35, normalized size = 0.53 \[ -\frac {1}{16} \, {\left (8 \, x^{7} + 36 \, x^{5} + 98 \, x^{3} + 147 \, x\right )} e^{\left (-x^{2}\right )} + \frac {147}{32} \, \sqrt {\pi } \operatorname {erf}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 51, normalized size = 0.77 \[ -\frac {x^{7} {\mathrm e}^{-x^{2}}}{2}-\frac {9 x^{5} {\mathrm e}^{-x^{2}}}{4}-\frac {49 x^{3} {\mathrm e}^{-x^{2}}}{8}-\frac {147 x \,{\mathrm e}^{-x^{2}}}{16}+\frac {147 \sqrt {\pi }\, \erf \relax (x )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 74, normalized size = 1.12 \[ -\frac {1}{16} \, {\left (8 \, x^{7} + 28 \, x^{5} + 70 \, x^{3} + 105 \, x\right )} e^{\left (-x^{2}\right )} - \frac {1}{8} \, {\left (4 \, x^{5} + 10 \, x^{3} + 15 \, x\right )} e^{\left (-x^{2}\right )} - \frac {1}{4} \, {\left (2 \, x^{3} + 3 \, x\right )} e^{\left (-x^{2}\right )} + \frac {147}{32} \, \sqrt {\pi } \operatorname {erf}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.62, size = 50, normalized size = 0.76 \[ \frac {147\,\sqrt {\pi }\,\mathrm {erf}\relax (x)}{32}-\frac {49\,x^3\,{\mathrm {e}}^{-x^2}}{8}-\frac {9\,x^5\,{\mathrm {e}}^{-x^2}}{4}-\frac {x^7\,{\mathrm {e}}^{-x^2}}{2}-\frac {147\,x\,{\mathrm {e}}^{-x^2}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 97.77, size = 54, normalized size = 0.82 \[ - \frac {x^{7} e^{- x^{2}}}{2} - \frac {9 x^{5} e^{- x^{2}}}{4} - \frac {49 x^{3} e^{- x^{2}}}{8} - \frac {147 x e^{- x^{2}}}{16} + \frac {147 \sqrt {\pi } \operatorname {erf}{\relax (x )}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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