Optimal. Leaf size=90 \[ \frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {x^2 e^{-b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {x e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]
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Rubi [A] time = 0.10, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6385, 6382, 2205, 2212} \[ -\frac {x^2 e^{-b^2 x^2} \text {Erf}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {Erf}(b x)}{2 b^4}+\frac {5 \text {Erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {x e^{-2 b^2 x^2}}{4 \sqrt {\pi } b^3} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2212
Rule 6382
Rule 6385
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {\int e^{-b^2 x^2} x \text {erf}(b x) \, dx}{b^2}+\frac {\int e^{-2 b^2 x^2} x^2 \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {\int e^{-2 b^2 x^2} \, dx}{4 b^3 \sqrt {\pi }}+\frac {\int e^{-2 b^2 x^2} \, dx}{b^3 \sqrt {\pi }}\\ &=-\frac {e^{-2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erf}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 68, normalized size = 0.76 \[ \frac {-8 e^{-b^2 x^2} \left (b^2 x^2+1\right ) \text {erf}(b x)-\frac {4 b x e^{-2 b^2 x^2}}{\sqrt {\pi }}+5 \sqrt {2} \text {erf}\left (\sqrt {2} b x\right )}{16 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 76, normalized size = 0.84 \[ -\frac {4 \, \sqrt {\pi } b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} - 5 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 8 \, {\left (\pi b^{3} x^{2} + \pi b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{16 \, \pi b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 94, normalized size = 1.04 \[ -\frac {{\left (b^{2} x^{2} + 1\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{4}} - \frac {b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {4 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{16 \, \sqrt {\pi } b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 83, normalized size = 0.92 \[ \frac {\frac {\erf \left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )}{16}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{4}}{b^{3} \sqrt {\pi }}}{b} \]
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 \[ -\frac {{\left (b^{2} x^{2} + 1\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, b^{4}} + \frac {-\frac {1}{16} \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{4 \, b}}{\sqrt {\pi } b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 106, normalized size = 1.18 \[ \frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{4\,b\,{\left (b^2\right )}^{3/2}}-\frac {\mathrm {erfi}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )}{4\,b\,{\left (-2\,b^2\right )}^{3/2}}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{2\,b^4}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}\right )-\frac {x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{4\,b^3\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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