Integrand size = 16, antiderivative size = 50 \[ \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx=\frac {e^c \sqrt {\pi } \text {erfi}(b x)}{2 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }} \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6512, 2235, 6511} \[ \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx=\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }} \]
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Rule 2235
Rule 6511
Rule 6512
Rubi steps \begin{align*} \text {integral}& = \int e^{c+b^2 x^2} \, dx-\int e^{c+b^2 x^2} \text {erf}(b x) \, dx \\ & = \frac {e^c \sqrt {\pi } \text {erfi}(b x)}{2 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}
\[ \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx=\int e^{c+b^2 x^2} \text {erfc}(b x) \, dx \]
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\[\int {\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right )d x\]
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\[ \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx=\int { \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
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Result contains complex when optimal does not.
Time = 4.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx=- \frac {b x^{2} e^{c} {{}_{2}F_{2}\left (\begin {matrix} 1, 1 \\ \frac {3}{2}, 2 \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi }} - \frac {i \sqrt {\pi } e^{c} \operatorname {erf}{\left (i b x \right )}}{2 b} \]
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\[ \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx=\int { \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
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\[ \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx=\int { \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \]
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Timed out. \[ \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx=\int {\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]
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