Integrand size = 18, antiderivative size = 21 \[ \int \frac {e^{c+b^2 x^2}}{\text {erfi}(b x)^2} \, dx=-\frac {e^c \sqrt {\pi }}{2 b \text {erfi}(b x)} \]
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Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6510, 30} \[ \int \frac {e^{c+b^2 x^2}}{\text {erfi}(b x)^2} \, dx=-\frac {\sqrt {\pi } e^c}{2 b \text {erfi}(b x)} \]
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Rule 30
Rule 6510
Rubi steps \begin{align*} \text {integral}& = \frac {\left (e^c \sqrt {\pi }\right ) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\text {erfi}(b x)\right )}{2 b} \\ & = -\frac {e^c \sqrt {\pi }}{2 b \text {erfi}(b x)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{c+b^2 x^2}}{\text {erfi}(b x)^2} \, dx=-\frac {e^c \sqrt {\pi }}{2 b \text {erfi}(b x)} \]
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\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c}}{\operatorname {erfi}\left (b x \right )^{2}}d x\]
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {e^{c+b^2 x^2}}{\text {erfi}(b x)^2} \, dx=-\frac {\sqrt {\pi } e^{c}}{2 \, b \operatorname {erfi}\left (b x\right )} \]
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Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {e^{c+b^2 x^2}}{\text {erfi}(b x)^2} \, dx=\begin {cases} - \frac {\sqrt {\pi } e^{c}}{2 b \operatorname {erfi}{\left (b x \right )}} & \text {for}\: b \neq 0 \\\tilde {\infty } x e^{c} & \text {otherwise} \end {cases} \]
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\[ \int \frac {e^{c+b^2 x^2}}{\text {erfi}(b x)^2} \, dx=\int { \frac {e^{\left (b^{2} x^{2} + c\right )}}{\operatorname {erfi}\left (b x\right )^{2}} \,d x } \]
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\[ \int \frac {e^{c+b^2 x^2}}{\text {erfi}(b x)^2} \, dx=\int { \frac {e^{\left (b^{2} x^{2} + c\right )}}{\operatorname {erfi}\left (b x\right )^{2}} \,d x } \]
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Time = 0.11 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {e^{c+b^2 x^2}}{\text {erfi}(b x)^2} \, dx=-\frac {\sqrt {\pi }\,{\mathrm {e}}^c}{2\,b\,\mathrm {erfi}\left (b\,x\right )} \]
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