Optimal. Leaf size=27 \[ x \text {erfc}(b x)-\frac {e^{-b^2 x^2}}{\sqrt {\pi } b} \]
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Rubi [A] time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6350} \[ x \text {Erfc}(b x)-\frac {e^{-b^2 x^2}}{\sqrt {\pi } b} \]
Antiderivative was successfully verified.
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Rule 6350
Rubi steps
\begin {align*} \int \text {erfc}(b x) \, dx &=-\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erfc}(b x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 27, normalized size = 1.00 \[ x \text {erfc}(b x)-\frac {e^{-b^2 x^2}}{\sqrt {\pi } b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 35, normalized size = 1.30 \[ -\frac {\pi b x \operatorname {erf}\left (b x\right ) - \pi b x + \sqrt {\pi } e^{\left (-b^{2} x^{2}\right )}}{\pi b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 26, normalized size = 0.96 \[ -x \operatorname {erf}\left (b x\right ) + x - \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi } b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 1.00 \[ \frac {b x \,\mathrm {erfc}\left (b x \right )-\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 26, normalized size = 0.96 \[ \frac {b x \operatorname {erfc}\left (b x\right ) - \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi }}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 24, normalized size = 0.89 \[ x\,\mathrm {erfc}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b^2\,x^2}}{b\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 24, normalized size = 0.89 \[ \begin {cases} x \operatorname {erfc}{\left (b x \right )} - \frac {e^{- b^{2} x^{2}}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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