Integrand size = 19, antiderivative size = 21 \[ \int e^{c-b^2 x^2} \text {erf}(b x)^2 \, dx=\frac {e^c \sqrt {\pi } \text {erf}(b x)^3}{6 b} \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6508, 30} \[ \int e^{c-b^2 x^2} \text {erf}(b x)^2 \, dx=\frac {\sqrt {\pi } e^c \text {erf}(b x)^3}{6 b} \]
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Rule 30
Rule 6508
Rubi steps \begin{align*} \text {integral}& = \frac {\left (e^c \sqrt {\pi }\right ) \text {Subst}\left (\int x^2 \, dx,x,\text {erf}(b x)\right )}{2 b} \\ & = \frac {e^c \sqrt {\pi } \text {erf}(b x)^3}{6 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{c-b^2 x^2} \text {erf}(b x)^2 \, dx=\frac {e^c \sqrt {\pi } \text {erf}(b x)^3}{6 b} \]
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Time = 0.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {{\mathrm e}^{c} \operatorname {erf}\left (b x \right )^{3} \sqrt {\pi }}{6 b}\) | \(17\) |
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none
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{c-b^2 x^2} \text {erf}(b x)^2 \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (b x\right )^{3} e^{c}}{6 \, b} \]
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Time = 0.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int e^{c-b^2 x^2} \text {erf}(b x)^2 \, dx=\begin {cases} \frac {\sqrt {\pi } e^{c} \operatorname {erf}^{3}{\left (b x \right )}}{6 b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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none
Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{c-b^2 x^2} \text {erf}(b x)^2 \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (b x\right )^{3} e^{c}}{6 \, b} \]
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\[ \int e^{c-b^2 x^2} \text {erf}(b x)^2 \, dx=\int { \operatorname {erf}\left (b x\right )^{2} e^{\left (-b^{2} x^{2} + c\right )} \,d x } \]
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Time = 5.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{c-b^2 x^2} \text {erf}(b x)^2 \, dx=\frac {\sqrt {\pi }\,{\mathrm {e}}^c\,{\mathrm {erf}\left (b\,x\right )}^3}{6\,b} \]
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