Optimal. Leaf size=27 \[ x \text{Erfc}(b x)-\frac{e^{-b^2 x^2}}{\sqrt{\pi } b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0049246, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6350} \[ x \text{Erfc}(b x)-\frac{e^{-b^2 x^2}}{\sqrt{\pi } b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6350
Rubi steps
\begin{align*} \int \text{erfc}(b x) \, dx &=-\frac{e^{-b^2 x^2}}{b \sqrt{\pi }}+x \text{erfc}(b x)\\ \end{align*}
Mathematica [A] time = 0.0021854, size = 27, normalized size = 1. \[ x \text{Erfc}(b x)-\frac{e^{-b^2 x^2}}{\sqrt{\pi } b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 27, normalized size = 1. \begin{align*}{\frac{1}{b} \left ( bx{\it erfc} \left ( bx \right ) -{\frac{{{\rm e}^{-{b}^{2}{x}^{2}}}}{\sqrt{\pi }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03127, size = 35, normalized size = 1.3 \begin{align*} \frac{b x \operatorname{erfc}\left (b x\right ) - \frac{e^{\left (-b^{2} x^{2}\right )}}{\sqrt{\pi }}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.31983, size = 81, normalized size = 3. \begin{align*} -\frac{\pi b x \operatorname{erf}\left (b x\right ) - \pi b x + \sqrt{\pi } e^{\left (-b^{2} x^{2}\right )}}{\pi b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.346812, size = 24, normalized size = 0.89 \begin{align*} \begin{cases} x \operatorname{erfc}{\left (b x \right )} - \frac{e^{- b^{2} x^{2}}}{\sqrt{\pi } b} & \text{for}\: b \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.32128, size = 35, normalized size = 1.3 \begin{align*} -x \operatorname{erf}\left (b x\right ) + x - \frac{e^{\left (-b^{2} x^{2}\right )}}{\sqrt{\pi } b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]