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 = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 12
Rule 2205
Rule 6353
Rule 6383
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*}
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Mathematica [A] time = 0.05, size = 56, normalized size = 1.00 \[ -\frac {2 e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } b}-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b}+x \text {erfc}(b x)^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 85, normalized size = 1.52 \[ \frac {\pi b^{2} x \operatorname {erf}\left (b x\right )^{2} - 2 \, \pi b^{2} x \operatorname {erf}\left (b x\right ) + \pi b^{2} x - \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 2 \, \sqrt {\pi } {\left (b \operatorname {erf}\left (b x\right ) - b\right )} e^{\left (-b^{2} x^{2}\right )}}{\pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 73, normalized size = 1.30 \[ x \operatorname {erf}\left (b x\right )^{2} - 2 \, x \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{2}}\right )}}{\sqrt {\pi }} + x - \frac {2 \, e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi } b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.86 \[ \frac {b x \erf \left (b x \right )^{2}+\frac {2 \erf \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \erf \left (b x \sqrt {2}\right )}{\sqrt {\pi }}}{b} \]
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 \[ \int \operatorname {erfc}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {erfc}\left (b\,x\right )}^2 \,d x \]
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 \[ \int \operatorname {erfc}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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