Optimal. Leaf size=20 \[ -\frac{\sqrt{\pi } e^c \log (\text{Erfc}(b x))}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02752, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {6374, 29} \[ -\frac{\sqrt{\pi } e^c \log (\text{Erfc}(b x))}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6374
Rule 29
Rubi steps
\begin{align*} \int \frac{e^{c-b^2 x^2}}{\text{erfc}(b x)} \, dx &=-\frac{\left (e^c \sqrt{\pi }\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\text{erfc}(b x)\right )}{2 b}\\ &=-\frac{e^c \sqrt{\pi } \log (\text{erfc}(b x))}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0107057, size = 20, normalized size = 1. \[ -\frac{\sqrt{\pi } e^c \log (\text{Erfc}(b x))}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{-{b}^{2}{x}^{2}+c}}}{{\it erfc} \left ( bx \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname{erfc}\left (b x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.07644, size = 53, normalized size = 2.65 \begin{align*} -\frac{\sqrt{\pi } e^{c} \log \left (\operatorname{erf}\left (b x\right ) - 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.548293, size = 24, normalized size = 1.2 \begin{align*} \begin{cases} - \frac{\sqrt{\pi } e^{c} \log{\left (\operatorname{erfc}{\left (b x \right )} \right )}}{2 b} & \text{for}\: b \neq 0 \\x e^{c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname{erfc}\left (b x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]