Optimal. Leaf size=109 \[ \frac {x^6 e^{-b^2 x^2}}{7 \sqrt {\pi } b}+\frac {6 e^{-b^2 x^2}}{7 \sqrt {\pi } b^7}+\frac {6 x^2 e^{-b^2 x^2}}{7 \sqrt {\pi } b^5}+\frac {3 x^4 e^{-b^2 x^2}}{7 \sqrt {\pi } b^3}+\frac {1}{7} x^7 \text {erf}(b x) \]
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Rubi [A] time = 0.10, 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 = {6361, 2212, 2209} \[ \frac {x^6 e^{-b^2 x^2}}{7 \sqrt {\pi } b}+\frac {3 x^4 e^{-b^2 x^2}}{7 \sqrt {\pi } b^3}+\frac {6 x^2 e^{-b^2 x^2}}{7 \sqrt {\pi } b^5}+\frac {6 e^{-b^2 x^2}}{7 \sqrt {\pi } b^7}+\frac {1}{7} x^7 \text {Erf}(b x) \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rule 6361
Rubi steps
\begin {align*} \int x^6 \text {erf}(b x) \, dx &=\frac {1}{7} x^7 \text {erf}(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 {erf}(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 {erf}(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 {erf}(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 {erf}(b x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 72, normalized size = 0.66 \[ \frac {e^{-b^2 x^2} \left (b^6 x^6+3 b^4 x^4+6 b^2 x^2+\sqrt {\pi } b^7 x^7 e^{b^2 x^2} \text {erf}(b x)+6\right )}{7 \sqrt {\pi } b^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 59, normalized size = 0.54 \[ \frac {\pi b^{7} x^{7} \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \pi b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 52, normalized size = 0.48 \[ \frac {1}{7} \, x^{7} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.83 \[ \frac {\frac {\erf \left (b x \right ) b^{7} x^{7}}{7}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{6} x^{6}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} b^{4} x^{4}}{2}-3 \,{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}-3 \,{\mathrm e}^{-b^{2} x^{2}}\right )}{7 \sqrt {\pi }}}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 52, normalized size = 0.48 \[ \frac {1}{7} \, x^{7} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{6} x^{6} + 3 \, b^{4} x^{4} + 6 \, b^{2} x^{2} + 6\right )} e^{\left (-b^{2} x^{2}\right )}}{7 \, \sqrt {\pi } b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 52, normalized size = 0.48 \[ \frac {x^7\,\mathrm {erf}\left (b\,x\right )}{7}+\frac {{\mathrm {e}}^{-b^2\,x^2}\,\left (b^6\,x^6+3\,b^4\,x^4+6\,b^2\,x^2+6\right )}{7\,b^7\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.16, size = 99, normalized size = 0.91 \[ \begin {cases} \frac {x^{7} \operatorname {erf}{\left (b x \right )}}{7} + \frac {x^{6} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b} + \frac {3 x^{4} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{3}} + \frac {6 x^{2} e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{5}} + \frac {6 e^{- b^{2} x^{2}}}{7 \sqrt {\pi } b^{7}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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