Optimal. Leaf size=112 \[ \frac {3 \sqrt {\pi } \text {erf}(b x)^2}{16 b^5}-\frac {x^3 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-2 b^2 x^2}}{2 \sqrt {\pi } b^5}-\frac {3 x e^{-b^2 x^2} \text {erf}(b x)}{4 b^4}-\frac {x^2 e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]
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Rubi [A] time = 0.15, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6385, 6373, 30, 2209, 2212} \[ -\frac {x^3 e^{-b^2 x^2} \text {Erf}(b x)}{2 b^2}-\frac {3 x e^{-b^2 x^2} \text {Erf}(b x)}{4 b^4}+\frac {3 \sqrt {\pi } \text {Erf}(b x)^2}{16 b^5}-\frac {x^2 e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3}-\frac {e^{-2 b^2 x^2}}{2 \sqrt {\pi } b^5} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2209
Rule 2212
Rule 6373
Rule 6385
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x^4 \text {erf}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 \int e^{-b^2 x^2} x^2 \text {erf}(b x) \, dx}{2 b^2}+\frac {\int e^{-2 b^2 x^2} x^3 \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 \int e^{-b^2 x^2} \text {erf}(b x) \, dx}{4 b^4}+\frac {\int e^{-2 b^2 x^2} x \, dx}{2 b^3 \sqrt {\pi }}+\frac {3 \int e^{-2 b^2 x^2} x \, dx}{2 b^3 \sqrt {\pi }}\\ &=-\frac {e^{-2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {\left (3 \sqrt {\pi }\right ) \operatorname {Subst}(\int x \, dx,x,\text {erf}(b x))}{8 b^5}\\ &=-\frac {e^{-2 b^2 x^2}}{2 b^5 \sqrt {\pi }}-\frac {e^{-2 b^2 x^2} x^2}{4 b^3 \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erf}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 \sqrt {\pi } \text {erf}(b x)^2}{16 b^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 85, normalized size = 0.76 \[ \frac {e^{-2 b^2 x^2} \left (3 \pi e^{2 b^2 x^2} \text {erf}(b x)^2-4 \sqrt {\pi } b x e^{b^2 x^2} \left (2 b^2 x^2+3\right ) \text {erf}(b x)-4 \left (b^2 x^2+2\right )\right )}{16 \sqrt {\pi } b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 74, normalized size = 0.66 \[ -\frac {4 \, {\left (2 \, \pi b^{3} x^{3} + 3 \, \pi b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (3 \, \pi \operatorname {erf}\left (b x\right )^{2} - 4 \, {\left (b^{2} x^{2} + 2\right )} e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{16 \, \pi b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int x^{4} \erf \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {-\frac {{\left (2 \, b^{2} x^{2} + 1\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}} - \frac {3 \, e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, b^{2}}}{2 \, \sqrt {\pi } b^{3}} - \frac {4 \, {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 3 \, \sqrt {\pi } \operatorname {erf}\left (b x\right )^{2}}{16 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 90, normalized size = 0.80 \[ -\frac {8\,{\mathrm {e}}^{-2\,b^2\,x^2}-3\,\pi \,{\mathrm {erf}\left (b\,x\right )}^2}{16\,b^5\,\sqrt {\pi }}-\frac {x^2\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4\,b^3\,\sqrt {\pi }}-\frac {3\,x\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{4\,b^4}-\frac {x^3\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.09, size = 109, normalized size = 0.97 \[ \begin {cases} - \frac {x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} - \frac {x^{2} e^{- 2 b^{2} x^{2}}}{4 \sqrt {\pi } b^{3}} - \frac {3 x e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{4 b^{4}} + \frac {3 \sqrt {\pi } \operatorname {erf}^{2}{\left (b x \right )}}{16 b^{5}} - \frac {e^{- 2 b^{2} x^{2}}}{2 \sqrt {\pi } b^{5}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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