Integrand size = 6, antiderivative size = 36 \[ \int \text {erf}(a+b x) \, dx=\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erf}(a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 36, 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 = {6484} \[ \int \text {erf}(a+b x) \, dx=\frac {(a+b x) \text {erf}(a+b x)}{b}+\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]
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Rule 6484
Rubi steps \begin{align*} \text {integral}& = \frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erf}(a+b x)}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \text {erf}(a+b x) \, dx=\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\left (\frac {a}{b}+x\right ) \text {erf}(a+b x) \]
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Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\operatorname {erf}\left (b x +a \right ) \left (b x +a \right )+\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(32\) |
default | \(\frac {\operatorname {erf}\left (b x +a \right ) \left (b x +a \right )+\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(32\) |
parallelrisch | \(\frac {x \,\operatorname {erf}\left (b x +a \right ) \sqrt {\pi }\, b +a \,\operatorname {erf}\left (b x +a \right ) \sqrt {\pi }+{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }\, b}\) | \(42\) |
parts | \(x \,\operatorname {erf}\left (b x +a \right )-\frac {2 b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 a b x -a^{2}}}{2 b^{2}}-\frac {a \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{\sqrt {\pi }}\) | \(57\) |
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Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31 \[ \int \text {erf}(a+b x) \, dx=\frac {{\left (\pi b x + \pi a\right )} \operatorname {erf}\left (b x + a\right ) + \sqrt {\pi } e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b} \]
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Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.47 \[ \int \text {erf}(a+b x) \, dx=\begin {cases} \frac {a \operatorname {erf}{\left (a + b x \right )}}{b} + x \operatorname {erf}{\left (a + b x \right )} + \frac {e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erf}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \text {erf}(a+b x) \, dx=\frac {{\left (b x + a\right )} \operatorname {erf}\left (b x + a\right ) + \frac {e^{\left (-{\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \]
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.64 \[ \int \text {erf}(a+b x) \, dx=x \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }} \]
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Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \text {erf}(a+b x) \, dx=x\,\mathrm {erf}\left (a+b\,x\right )+\frac {a\,\mathrm {erf}\left (a+b\,x\right )}{b}+\frac {{\mathrm {e}}^{-b^2\,x^2}\,{\mathrm {e}}^{-a^2}\,{\mathrm {e}}^{-2\,a\,b\,x}}{b\,\sqrt {\pi }} \]
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