Optimal. Leaf size=71 \[ \frac {3 e^{-b^2 x^2} x}{8 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^3}{4 b \sqrt {\pi }}-\frac {3 \text {Erf}(b x)}{16 b^4}+\frac {1}{4} x^4 \text {Erf}(b x) \]
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Rubi [A]
time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2243,
2236} \begin {gather*} -\frac {3 \text {Erf}(b x)}{16 b^4}+\frac {x^3 e^{-b^2 x^2}}{4 \sqrt {\pi } b}+\frac {3 x e^{-b^2 x^2}}{8 \sqrt {\pi } b^3}+\frac {1}{4} x^4 \text {Erf}(b x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2236
Rule 2243
Rule 6496
Rubi steps
\begin {align*} \int x^3 \text {erf}(b x) \, dx &=\frac {1}{4} x^4 \text {erf}(b x)-\frac {b \int e^{-b^2 x^2} x^4 \, dx}{2 \sqrt {\pi }}\\ &=\frac {e^{-b^2 x^2} x^3}{4 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erf}(b x)-\frac {3 \int e^{-b^2 x^2} x^2 \, dx}{4 b \sqrt {\pi }}\\ &=\frac {3 e^{-b^2 x^2} x}{8 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^3}{4 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erf}(b x)-\frac {3 \int e^{-b^2 x^2} \, dx}{8 b^3 \sqrt {\pi }}\\ &=\frac {3 e^{-b^2 x^2} x}{8 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^3}{4 b \sqrt {\pi }}-\frac {3 \text {erf}(b x)}{16 b^4}+\frac {1}{4} x^4 \text {erf}(b x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 63, normalized size = 0.89 \begin {gather*} e^{-b^2 x^2} \left (\frac {3 x}{8 b^3 \sqrt {\pi }}+\frac {x^3}{4 b \sqrt {\pi }}\right )-\frac {3 \text {Erf}(b x)}{16 b^4}+\frac {1}{4} x^4 \text {Erf}(b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 65, normalized size = 0.92
method | result | size |
meijerg | \(\frac {\frac {x b \left (10 b^{2} x^{2}+15\right ) {\mathrm e}^{-b^{2} x^{2}}}{20}-\frac {\left (-20 b^{4} x^{4}+15\right ) \erf \left (b x \right ) \sqrt {\pi }}{40}}{2 b^{4} \sqrt {\pi }}\) | \(52\) |
derivativedivides | \(\frac {\frac {\erf \left (b x \right ) b^{4} x^{4}}{4}-\frac {-\frac {b^{3} x^{3} {\mathrm e}^{-b^{2} x^{2}}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} b x}{4}+\frac {3 \erf \left (b x \right ) \sqrt {\pi }}{8}}{2 \sqrt {\pi }}}{b^{4}}\) | \(65\) |
default | \(\frac {\frac {\erf \left (b x \right ) b^{4} x^{4}}{4}-\frac {-\frac {b^{3} x^{3} {\mathrm e}^{-b^{2} x^{2}}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} b x}{4}+\frac {3 \erf \left (b x \right ) \sqrt {\pi }}{8}}{2 \sqrt {\pi }}}{b^{4}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 55, normalized size = 0.77 \begin {gather*} \frac {1}{4} \, x^{4} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (2 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{4}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (b x\right )}{b^{5}}\right )}}{16 \, \sqrt {\pi }} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 55, normalized size = 0.77 \begin {gather*} \frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )}{16 \, \pi b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 65, normalized size = 0.92 \begin {gather*} \begin {cases} \frac {x^{4} \operatorname {erf}{\left (b x \right )}}{4} + \frac {x^{3} e^{- b^{2} x^{2}}}{4 \sqrt {\pi } b} + \frac {3 x e^{- b^{2} x^{2}}}{8 \sqrt {\pi } b^{3}} - \frac {3 \operatorname {erf}{\left (b x \right )}}{16 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 56, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, x^{4} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (2 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-b x\right )}{b^{5}}\right )}}{16 \, \sqrt {\pi }} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 88, normalized size = 1.24 \begin {gather*} \frac {x^4\,\mathrm {erf}\left (b\,x\right )}{4}-\frac {3\,b\,x^5}{16\,{\left (b^2\,x^2\right )}^{5/2}}+\frac {x^3\,{\mathrm {e}}^{-b^2\,x^2}}{4\,b\,\sqrt {\pi }}+\frac {3\,x\,{\mathrm {e}}^{-b^2\,x^2}}{8\,b^3\,\sqrt {\pi }}+\frac {3\,b\,x^5\,\mathrm {erfc}\left (\sqrt {b^2\,x^2}\right )}{16\,{\left (b^2\,x^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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