Optimal. Leaf size=37 \[ \frac {(a+b x) \text {erfc}(a+b x)}{b}-\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6350} \[ \frac {(a+b x) \text {Erfc}(a+b x)}{b}-\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6350
Rubi steps
\begin {align*} \int \text {erfc}(a+b x) \, dx &=-\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 42, normalized size = 1.14 \[ -\frac {a \text {erf}(a+b x)}{b}+x \text {erfc}(a+b x)-\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 53, normalized size = 1.43 \[ \frac {\pi b x - {\left (\pi b x + \pi a\right )} \operatorname {erf}\left (b x + a\right ) - \sqrt {\pi } e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.40, size = 60, normalized size = 1.62 \[ -x \operatorname {erf}\left (b x + a\right ) + x + \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 33, normalized size = 0.89 \[ \frac {\left (b x +a \right ) \mathrm {erfc}\left (b x +a \right )-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.13, size = 32, normalized size = 0.86 \[ \frac {{\left (b x + a\right )} \operatorname {erfc}\left (b x + a\right ) - \frac {e^{\left (-{\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 49, normalized size = 1.32 \[ x\,\mathrm {erfc}\left (a+b\,x\right )+\frac {a\,\mathrm {erfc}\left (a+b\,x\right )}{b}-\frac {{\mathrm {e}}^{-b^2\,x^2}\,{\mathrm {e}}^{-a^2}\,{\mathrm {e}}^{-2\,a\,b\,x}}{b\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.61, size = 53, normalized size = 1.43 \[ \begin {cases} \frac {a \operatorname {erfc}{\left (a + b x \right )}}{b} + x \operatorname {erfc}{\left (a + b x \right )} - \frac {e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erfc}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________