Optimal. Leaf size=72 \[ -\frac {x e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } b}-\frac {\text {erfc}(b x)^2}{4 b^2}+\frac {e^{-2 b^2 x^2}}{2 \pi b^2}+\frac {1}{2} x^2 \text {erfc}(b x)^2 \]
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Rubi [A] time = 0.08, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6365, 6386, 6374, 30, 2209} \[ -\frac {x e^{-b^2 x^2} \text {Erfc}(b x)}{\sqrt {\pi } b}-\frac {\text {Erfc}(b x)^2}{4 b^2}+\frac {e^{-2 b^2 x^2}}{2 \pi b^2}+\frac {1}{2} x^2 \text {Erfc}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 2209
Rule 6365
Rule 6374
Rule 6386
Rubi steps
\begin {align*} \int x \text {erfc}(b x)^2 \, dx &=\frac {1}{2} x^2 \text {erfc}(b x)^2+\frac {(2 b) \int e^{-b^2 x^2} x^2 \text {erfc}(b x) \, dx}{\sqrt {\pi }}\\ &=-\frac {e^{-b^2 x^2} x \text {erfc}(b x)}{b \sqrt {\pi }}+\frac {1}{2} x^2 \text {erfc}(b x)^2-\frac {2 \int e^{-2 b^2 x^2} x \, dx}{\pi }+\frac {\int e^{-b^2 x^2} \text {erfc}(b x) \, dx}{b \sqrt {\pi }}\\ &=\frac {e^{-2 b^2 x^2}}{2 b^2 \pi }-\frac {e^{-b^2 x^2} x \text {erfc}(b x)}{b \sqrt {\pi }}+\frac {1}{2} x^2 \text {erfc}(b x)^2-\frac {\operatorname {Subst}(\int x \, dx,x,\text {erfc}(b x))}{2 b^2}\\ &=\frac {e^{-2 b^2 x^2}}{2 b^2 \pi }-\frac {e^{-b^2 x^2} x \text {erfc}(b x)}{b \sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{4 b^2}+\frac {1}{2} x^2 \text {erfc}(b x)^2\\ \end {align*}
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Mathematica [A] time = 0.16, size = 99, normalized size = 1.38 \[ \frac {\pi \left (2 b^2 x^2-1\right ) \text {erf}(b x)^2+\left (4 \sqrt {\pi } b x e^{-b^2 x^2}+\pi \left (2-4 b^2 x^2\right )\right ) \text {erf}(b x)+2 e^{-2 b^2 x^2} \left (\sqrt {\pi } b x e^{b^2 x^2}-1\right )^2}{4 \pi b^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 91, normalized size = 1.26 \[ \frac {2 \, \pi b^{2} x^{2} - {\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, \sqrt {\pi } {\left (b x \operatorname {erf}\left (b x\right ) - b x\right )} e^{\left (-b^{2} x^{2}\right )} + 2 \, {\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right ) + 2 \, e^{\left (-2 \, b^{2} x^{2}\right )}}{4 \, \pi b^{2}} \]
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 \[ \int x \operatorname {erfc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x \mathrm {erfc}\left (b x \right )^{2}\, dx \]
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 \[ \int x \operatorname {erfc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 68, normalized size = 0.94 \[ \frac {\frac {{\mathrm {e}}^{-2\,b^2\,x^2}}{2}-b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{b^2\,\pi }-\frac {\frac {{\mathrm {erfc}\left (b\,x\right )}^2}{4}-\frac {b^2\,x^2\,{\mathrm {erfc}\left (b\,x\right )}^2}{2}}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 68, normalized size = 0.94 \[ \begin {cases} \frac {x^{2} \operatorname {erfc}^{2}{\left (b x \right )}}{2} - \frac {x e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{\sqrt {\pi } b} - \frac {\operatorname {erfc}^{2}{\left (b x \right )}}{4 b^{2}} + \frac {e^{- 2 b^{2} x^{2}}}{2 \pi b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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