Optimal. Leaf size=71 \[ \frac {x}{\sqrt {\pi } b^3}-\frac {x^2 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^4}+\frac {x^3}{3 \sqrt {\pi } b} \]
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Rubi [A] time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6387, 6384, 8, 30} \[ -\frac {x^2 e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^2}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^4}+\frac {x}{\sqrt {\pi } b^3}+\frac {x^3}{3 \sqrt {\pi } b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 6384
Rule 6387
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x^3 \text {erfi}(b x) \, dx &=-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}+\frac {\int e^{-b^2 x^2} x \text {erfi}(b x) \, dx}{b^2}+\frac {\int x^2 \, dx}{b \sqrt {\pi }}\\ &=\frac {x^3}{3 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}+\frac {\int 1 \, dx}{b^3 \sqrt {\pi }}\\ &=\frac {x}{b^3 \sqrt {\pi }}+\frac {x^3}{3 b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^4}-\frac {e^{-b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.72 \[ \frac {\frac {2 b x \left (b^2 x^2+3\right )}{\sqrt {\pi }}-3 e^{-b^2 x^2} \left (b^2 x^2+1\right ) \text {erfi}(b x)}{6 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 59, normalized size = 0.83 \[ \frac {{\left (2 \, \sqrt {\pi } {\left (b^{3} x^{3} + 3 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - 3 \, {\left (\pi + \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{6 \, \pi b^{4}} \]
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 \[ \int x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 72, normalized size = 1.01 \[ \frac {\left (2 \,{\mathrm e}^{b^{2} x^{2}} b^{3} x^{3}-3 \sqrt {\pi }\, \erfi \left (b x \right ) b^{2} x^{2}+6 \,{\mathrm e}^{b^{2} x^{2}} b x -3 \sqrt {\pi }\, \erfi \left (b x \right )\right ) {\mathrm e}^{-b^{2} x^{2}}}{6 b^{4} \sqrt {\pi }} \]
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 \[ \int x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 56, normalized size = 0.79 \[ \frac {\frac {b^2\,x^3}{3}+x}{b^3\,\sqrt {\pi }}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{-b^2\,x^2}}{2\,b^4}+\frac {x^2\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 154.66, size = 63, normalized size = 0.89 \[ \begin {cases} \frac {x^{3}}{3 \sqrt {\pi } b} - \frac {x^{2} e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} + \frac {x}{\sqrt {\pi } b^{3}} - \frac {e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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