Integrand size = 17, antiderivative size = 86 \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {b e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \]
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Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6518, 2266, 2236} \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \sqrt {b^2-d}}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \]
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Rule 2236
Rule 2266
Rule 6518
Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}+\frac {b \int e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} \, dx}{d \sqrt {\pi }} \\ & = \frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}+\frac {\left (b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{d \sqrt {\pi }} \\ & = \frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {e^c \left (e^{d x^2} \text {erfc}(a+b x)+\frac {b e^{\frac {a^2 d}{b^2-d}} \text {erfi}\left (\frac {-a b+\left (-b^2+d\right ) x}{\sqrt {-b^2+d}}\right )}{\sqrt {-b^2+d}}\right )}{2 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(76)=152\).
Time = 0.90 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.08
method | result | size |
default | \(\frac {\frac {b \,{\mathrm e}^{\frac {a^{2} d -2 d a \left (b x +a \right )+b^{2} c +d \left (b x +a \right )^{2}}{b^{2}}}}{2 d}-\frac {\operatorname {erf}\left (b x +a \right ) b \,{\mathrm e}^{\frac {a^{2} d -2 d a \left (b x +a \right )+b^{2} c +d \left (b x +a \right )^{2}}{b^{2}}}}{2 d}+\frac {b \,{\mathrm e}^{\frac {a^{2} d +b^{2} c}{b^{2}}-\frac {a^{2} d^{2}}{b^{4} \left (-1+\frac {d}{b^{2}}\right )}} \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, \left (b x +a \right )+\frac {a d}{b^{2} \sqrt {1-\frac {d}{b^{2}}}}\right )}{2 d \sqrt {1-\frac {d}{b^{2}}}}}{b}\) | \(179\) |
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Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.26 \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\frac {\sqrt {b^{2} - d} b \operatorname {erf}\left (\frac {a b + {\left (b^{2} - d\right )} x}{\sqrt {b^{2} - d}}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} - c\right )} d}{b^{2} - d}\right )} + {\left (b^{2} - {\left (b^{2} - d\right )} \operatorname {erf}\left (b x + a\right ) - d\right )} e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d - d^{2}\right )}} \]
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\[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=e^{c} \int x e^{d x^{2}} \operatorname {erfc}{\left (a + b x \right )}\, dx \]
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\[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\int { x \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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\[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\int { x \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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Timed out. \[ \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx=\int x\,\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \]
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