Optimal. Leaf size=109 \[ -\frac {x^6 e^{-b^2 x^2}}{7 \sqrt {\pi } b}-\frac {6 e^{-b^2 x^2}}{7 \sqrt {\pi } b^7}-\frac {6 x^2 e^{-b^2 x^2}}{7 \sqrt {\pi } b^5}-\frac {3 x^4 e^{-b^2 x^2}}{7 \sqrt {\pi } b^3}+\frac {1}{7} x^7 \text {erfc}(b x) \]
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Rubi [A] time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6362, 2212, 2209} \[ -\frac {x^6 e^{-b^2 x^2}}{7 \sqrt {\pi } b}-\frac {3 x^4 e^{-b^2 x^2}}{7 \sqrt {\pi } b^3}-\frac {6 x^2 e^{-b^2 x^2}}{7 \sqrt {\pi } b^5}-\frac {6 e^{-b^2 x^2}}{7 \sqrt {\pi } b^7}+\frac {1}{7} x^7 \text {Erfc}(b x) \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rule 6362
Rubi steps
\begin {align*} \int x^6 \text {erfc}(b x) \, dx &=\frac {1}{7} x^7 \text {erfc}(b x)+\frac {(2 b) \int e^{-b^2 x^2} x^7 \, dx}{7 \sqrt {\pi }}\\ &=-\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfc}(b x)+\frac {6 \int e^{-b^2 x^2} x^5 \, dx}{7 b \sqrt {\pi }}\\ &=-\frac {3 e^{-b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfc}(b x)+\frac {12 \int e^{-b^2 x^2} x^3 \, dx}{7 b^3 \sqrt {\pi }}\\ &=-\frac {6 e^{-b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfc}(b x)+\frac {12 \int e^{-b^2 x^2} x \, dx}{7 b^5 \sqrt {\pi }}\\ &=-\frac {6 e^{-b^2 x^2}}{7 b^7 \sqrt {\pi }}-\frac {6 e^{-b^2 x^2} x^2}{7 b^5 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x^4}{7 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} x^6}{7 b \sqrt {\pi }}+\frac {1}{7} x^7 \text {erfc}(b x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 73, normalized size = 0.67 \[ \frac {e^{-b^2 x^2} \left (-b^6 x^6-3 b^4 x^4-6 b^2 x^2+\sqrt {\pi } b^7 x^7 e^{b^2 x^2} \text {erfc}(b x)-6\right )}{7 \sqrt {\pi } b^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 68, normalized size = 0.62 \[ -\frac {\pi b^{7} x^{7} \operatorname {erf}\left (b x\right ) - \pi b^{7} x^{7} + \sqrt {\pi } {\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \pi b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 57, normalized size = 0.52 \[ -\frac {1}{7} \, x^{7} \operatorname {erf}\left (b x\right ) + \frac {1}{7} \, x^{7} - \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 90, normalized size = 0.83 \[ \frac {\frac {b^{7} x^{7} \mathrm {erfc}\left (b x \right )}{7}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{6} x^{6}}{7}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} b^{4} x^{4}}{7}-\frac {6 \,{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}}{7}-\frac {6 \,{\mathrm e}^{-b^{2} x^{2}}}{7}}{\sqrt {\pi }}}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 52, normalized size = 0.48 \[ \frac {1}{7} \, x^{7} \operatorname {erfc}\left (b x\right ) - \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 90, normalized size = 0.83 \[ \frac {x^7\,\mathrm {erfc}\left (b\,x\right )}{7}-\frac {\frac {6\,{\mathrm {e}}^{-b^2\,x^2}}{7\,\sqrt {\pi }}+\frac {6\,b^2\,x^2\,{\mathrm {e}}^{-b^2\,x^2}}{7\,\sqrt {\pi }}+\frac {3\,b^4\,x^4\,{\mathrm {e}}^{-b^2\,x^2}}{7\,\sqrt {\pi }}+\frac {b^6\,x^6\,{\mathrm {e}}^{-b^2\,x^2}}{7\,\sqrt {\pi }}}{b^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.63, size = 102, normalized size = 0.94 \[ \begin {cases} \frac {x^{7} \operatorname {erfc}{\left (b x \right )}}{7} - \frac {x^{6} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b} - \frac {3 x^{4} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{3}} - \frac {6 x^{2} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{5}} - \frac {6 e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{7}} & \text {for}\: b \neq 0 \\\frac {x^{7}}{7} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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