Integrand size = 17, antiderivative size = 21 \[ \int e^{c-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {e^c \sqrt {\pi } \text {erfc}(b x)^2}{4 b} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6509, 30} \[ \int e^{c-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {\sqrt {\pi } e^c \text {erfc}(b x)^2}{4 b} \]
[In]
[Out]
Rule 30
Rule 6509
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (e^c \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfc}(b x))}{2 b} \\ & = -\frac {e^c \sqrt {\pi } \text {erfc}(b x)^2}{4 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{c-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {e^c \sqrt {\pi } \text {erfc}(b x)^2}{4 b} \]
[In]
[Out]
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43
method | result | size |
default | \(\frac {\frac {{\mathrm e}^{c} \sqrt {\pi }\, \operatorname {erf}\left (b x \right )}{2}-\frac {\sqrt {\pi }\, {\mathrm e}^{c} \operatorname {erf}\left (b x \right )^{2}}{4}}{b}\) | \(30\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int e^{c-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {\sqrt {\pi } {\left (\operatorname {erf}\left (b x\right )^{2} - 2 \, \operatorname {erf}\left (b x\right )\right )} e^{c}}{4 \, b} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int e^{c-b^2 x^2} \text {erfc}(b x) \, dx=\begin {cases} - \frac {\sqrt {\pi } e^{c} \operatorname {erfc}^{2}{\left (b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x e^{c} & \text {otherwise} \end {cases} \]
[In]
[Out]
\[ \int e^{c-b^2 x^2} \text {erfc}(b x) \, dx=\int { \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2} + c\right )} \,d x } \]
[In]
[Out]
\[ \int e^{c-b^2 x^2} \text {erfc}(b x) \, dx=\int { \operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2} + c\right )} \,d x } \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{c-b^2 x^2} \text {erfc}(b x) \, dx=-\frac {\sqrt {\pi }\,{\mathrm {e}}^c\,{\mathrm {erfc}\left (b\,x\right )}^2}{4\,b} \]
[In]
[Out]