Integrand size = 6, antiderivative size = 37 \[ \int \text {erfc}(a+b x) \, dx=-\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6485} \[ \int \text {erfc}(a+b x) \, dx=\frac {(a+b x) \text {erfc}(a+b x)}{b}-\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]
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Rule 6485
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)}{b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \text {erfc}(a+b x) \, dx=-\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}-\frac {a \text {erf}(a+b x)}{b}+x \text {erfc}(a+b x) \]
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Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\left (b x +a \right ) \operatorname {erfc}\left (b x +a \right )-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(33\) |
default | \(\frac {\left (b x +a \right ) \operatorname {erfc}\left (b x +a \right )-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(33\) |
parallelrisch | \(\frac {x \,\operatorname {erfc}\left (b x +a \right ) \sqrt {\pi }\, b +a \,\operatorname {erfc}\left (b x +a \right ) \sqrt {\pi }-{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }\, b}\) | \(44\) |
parts | \(x \,\operatorname {erfc}\left (b x +a \right )+\frac {2 b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x -a^{2}}}{2 b^{2}}-\frac {a \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{\sqrt {\pi }}\) | \(57\) |
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Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43 \[ \int \text {erfc}(a+b x) \, dx=\frac {\pi b x - {\left (\pi b x + \pi a\right )} \operatorname {erf}\left (b x + a\right ) - \sqrt {\pi } e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b} \]
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Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43 \[ \int \text {erfc}(a+b x) \, dx=\begin {cases} \frac {a \operatorname {erfc}{\left (a + b x \right )}}{b} + x \operatorname {erfc}{\left (a + b x \right )} - \frac {e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erfc}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \text {erfc}(a+b x) \, dx=\frac {{\left (b x + a\right )} \operatorname {erfc}\left (b x + a\right ) - \frac {e^{\left (-{\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \]
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.62 \[ \int \text {erfc}(a+b x) \, dx=-x \operatorname {erf}\left (b x + a\right ) + x + \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }} \]
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Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int \text {erfc}(a+b x) \, dx=x\,\mathrm {erfc}\left (a+b\,x\right )+\frac {a\,\mathrm {erfc}\left (a+b\,x\right )}{b}-\frac {{\mathrm {e}}^{-b^2\,x^2}\,{\mathrm {e}}^{-a^2}\,{\mathrm {e}}^{-2\,a\,b\,x}}{b\,\sqrt {\pi }} \]
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