Integrand size = 4, antiderivative size = 27 \[ \int \text {erfc}(b x) \, dx=-\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erfc}(b x) \]
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Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6485} \[ \int \text {erfc}(b x) \, dx=x \text {erfc}(b x)-\frac {e^{-b^2 x^2}}{\sqrt {\pi } b} \]
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Rule 6485
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erfc}(b x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \text {erfc}(b x) \, dx=-\frac {e^{-b^2 x^2}}{b \sqrt {\pi }}+x \text {erfc}(b x) \]
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Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
parts | \(x \,\operatorname {erfc}\left (b x \right )-\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }\, b}\) | \(25\) |
derivativedivides | \(\frac {b x \,\operatorname {erfc}\left (b x \right )-\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}}{b}\) | \(27\) |
default | \(\frac {b x \,\operatorname {erfc}\left (b x \right )-\frac {{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}}{b}\) | \(27\) |
parallelrisch | \(\frac {x \,\operatorname {erfc}\left (b x \right ) \sqrt {\pi }\, b -{\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }\, b}\) | \(30\) |
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Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \text {erfc}(b x) \, dx=-\frac {\pi b x \operatorname {erf}\left (b x\right ) - \pi b x + \sqrt {\pi } e^{\left (-b^{2} x^{2}\right )}}{\pi b} \]
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Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \text {erfc}(b x) \, dx=\begin {cases} x \operatorname {erfc}{\left (b x \right )} - \frac {e^{- b^{2} x^{2}}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \text {erfc}(b x) \, dx=\frac {b x \operatorname {erfc}\left (b x\right ) - \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi }}}{b} \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \text {erfc}(b x) \, dx=-x \operatorname {erf}\left (b x\right ) + x - \frac {e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi } b} \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \text {erfc}(b x) \, dx=x\,\mathrm {erfc}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b^2\,x^2}}{b\,\sqrt {\pi }} \]
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