Optimal. Leaf size=56 \[ -\frac{2 e^{-b^2 x^2} \text{Erfc}(b x)}{\sqrt{\pi } b}-\frac{\sqrt{\frac{2}{\pi }} \text{Erf}\left (\sqrt{2} b x\right )}{b}+x \text{Erfc}(b x)^2 \]
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Rubi [A] time = 0.0458552, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6353, 12, 6383, 2205} \[ -\frac{2 e^{-b^2 x^2} \text{Erfc}(b x)}{\sqrt{\pi } b}-\frac{\sqrt{\frac{2}{\pi }} \text{Erf}\left (\sqrt{2} b x\right )}{b}+x \text{Erfc}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6353
Rule 12
Rule 6383
Rule 2205
Rubi steps
\begin{align*} \int \text{erfc}(b x)^2 \, dx &=x \text{erfc}(b x)^2+\frac{4 \int b e^{-b^2 x^2} x \text{erfc}(b x) \, dx}{\sqrt{\pi }}\\ &=x \text{erfc}(b x)^2+\frac{(4 b) \int e^{-b^2 x^2} x \text{erfc}(b x) \, dx}{\sqrt{\pi }}\\ &=-\frac{2 e^{-b^2 x^2} \text{erfc}(b x)}{b \sqrt{\pi }}+x \text{erfc}(b x)^2-\frac{4 \int e^{-2 b^2 x^2} \, dx}{\pi }\\ &=-\frac{\sqrt{\frac{2}{\pi }} \text{erf}\left (\sqrt{2} b x\right )}{b}-\frac{2 e^{-b^2 x^2} \text{erfc}(b x)}{b \sqrt{\pi }}+x \text{erfc}(b x)^2\\ \end{align*}
Mathematica [A] time = 0.0497155, size = 56, normalized size = 1. \[ -\frac{2 e^{-b^2 x^2} \text{Erfc}(b x)}{\sqrt{\pi } b}-\frac{\sqrt{\frac{2}{\pi }} \text{Erf}\left (\sqrt{2} b x\right )}{b}+x \text{Erfc}(b x)^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 48, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( bx \left ({\it Erf} \left ( bx \right ) \right ) ^{2}+2\,{\frac{{\it Erf} \left ( bx \right ){{\rm e}^{-{b}^{2}{x}^{2}}}}{\sqrt{\pi }}}-{\frac{\sqrt{2}{\it Erf} \left ( bx\sqrt{2} \right ) }{\sqrt{\pi }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfc}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12992, size = 219, normalized size = 3.91 \begin{align*} \frac{\pi b^{2} x \operatorname{erf}\left (b x\right )^{2} - 2 \, \pi b^{2} x \operatorname{erf}\left (b x\right ) + \pi b^{2} x - \sqrt{2} \sqrt{\pi } \sqrt{b^{2}} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right ) + 2 \, \sqrt{\pi }{\left (b \operatorname{erf}\left (b x\right ) - b\right )} e^{\left (-b^{2} x^{2}\right )}}{\pi b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfc}^{2}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31717, size = 99, normalized size = 1.77 \begin{align*} x \operatorname{erf}\left (b x\right )^{2} - 2 \, x \operatorname{erf}\left (b x\right ) + \frac{b{\left (\frac{2 \, \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{2}} + \frac{\sqrt{2} \operatorname{erf}\left (-\sqrt{2} b x\right )}{b^{2}}\right )}}{\sqrt{\pi }} + x - \frac{2 \, e^{\left (-b^{2} x^{2}\right )}}{\sqrt{\pi } b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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