Optimal. Leaf size=113 \[ -\frac {5 \text {erf}\left (\sqrt {2} b x\right )}{6 \sqrt {2 \pi } b^3}+\frac {2 x^2 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } b}+\frac {x e^{-2 b^2 x^2}}{3 \pi b^2}+\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } b^3}+\frac {1}{3} x^3 \text {erf}(b x)^2 \]
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Rubi [A] time = 0.14, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6364, 6385, 6382, 2205, 2212} \[ \frac {2 x^2 e^{-b^2 x^2} \text {Erf}(b x)}{3 \sqrt {\pi } b}+\frac {2 e^{-b^2 x^2} \text {Erf}(b x)}{3 \sqrt {\pi } b^3}-\frac {5 \text {Erf}\left (\sqrt {2} b x\right )}{6 \sqrt {2 \pi } b^3}+\frac {x e^{-2 b^2 x^2}}{3 \pi b^2}+\frac {1}{3} x^3 \text {Erf}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2212
Rule 6364
Rule 6382
Rule 6385
Rubi steps
\begin {align*} \int x^2 \text {erf}(b x)^2 \, dx &=\frac {1}{3} x^3 \text {erf}(b x)^2-\frac {(4 b) \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx}{3 \sqrt {\pi }}\\ &=\frac {2 e^{-b^2 x^2} x^2 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erf}(b x)^2-\frac {4 \int e^{-2 b^2 x^2} x^2 \, dx}{3 \pi }-\frac {4 \int e^{-b^2 x^2} x \text {erf}(b x) \, dx}{3 b \sqrt {\pi }}\\ &=\frac {e^{-2 b^2 x^2} x}{3 b^2 \pi }+\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{3 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^2 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erf}(b x)^2-\frac {\int e^{-2 b^2 x^2} \, dx}{3 b^2 \pi }-\frac {4 \int e^{-2 b^2 x^2} \, dx}{3 b^2 \pi }\\ &=\frac {e^{-2 b^2 x^2} x}{3 b^2 \pi }+\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{3 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^2 \text {erf}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erf}(b x)^2-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{3 b^3}-\frac {\text {erf}\left (\sqrt {2} b x\right )}{6 b^3 \sqrt {2 \pi }}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 88, normalized size = 0.78 \[ \frac {4 \pi b^3 x^3 \text {erf}(b x)^2+8 \sqrt {\pi } e^{-b^2 x^2} \left (b^2 x^2+1\right ) \text {erf}(b x)+4 b x e^{-2 b^2 x^2}-5 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} b x\right )}{12 \pi b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 90, normalized size = 0.80 \[ \frac {4 \, \pi b^{4} x^{3} \operatorname {erf}\left (b x\right )^{2} + 4 \, b^{2} x e^{\left (-2 \, b^{2} x^{2}\right )} + 8 \, \sqrt {\pi } {\left (b^{3} x^{2} + b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 5 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{12 \, \pi b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 111, normalized size = 0.98 \[ \frac {1}{3} \, x^{3} \operatorname {erf}\left (b x\right )^{2} + \frac {b {\left (\frac {8 \, {\left (b^{2} x^{2} + 1\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{4}} + \frac {b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {4 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{\sqrt {\pi } b^{3}}\right )}}{12 \, \sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 95, normalized size = 0.84 \[ \frac {\frac {b^{3} x^{3} \erf \left (b x \right )^{2}}{3}-\frac {4 \erf \left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} b^{2} x^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}+\frac {-\frac {5 \sqrt {2}\, \sqrt {\pi }\, \erf \left (b x \sqrt {2}\right )}{12}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b x}{3}}{\pi }}{b^{3}} \]
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 \[ -\frac {-\frac {1}{4} \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} b x\right )}{b}}{3 \, \pi b^{2}} + \frac {\pi b^{3} x^{3} \operatorname {erf}\left (b x\right )^{2} + 2 \, {\left (\sqrt {\pi } b^{2} x^{2} + \sqrt {\pi }\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{3 \, \pi b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 90, normalized size = 0.80 \[ \frac {x^3\,{\mathrm {erf}\left (b\,x\right )}^2}{3}+\frac {\frac {2\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{3}-\frac {5\,\sqrt {2}\,\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {2}\,b\,x\right )}{12}+\frac {b\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{3}+\frac {2\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{3}}{b^3\,\pi } \]
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 \[ \int x^{2} \operatorname {erf}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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