Optimal. Leaf size=36 \[ \frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {Erf}(a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 36, 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 = {6484}
\begin {gather*} \frac {(a+b x) \text {Erf}(a+b x)}{b}+\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6484
Rubi steps
\begin {align*} \int \text {erf}(a+b x) \, dx &=\frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erf}(a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 35, normalized size = 0.97 \begin {gather*} \frac {e^{-(a+b x)^2}}{b \sqrt {\pi }}+\left (\frac {a}{b}+x\right ) \text {Erf}(a+b x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.25, size = 32, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\erf \left (b x +a \right ) \left (b x +a \right )+\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(32\) |
default | \(\frac {\erf \left (b x +a \right ) \left (b x +a \right )+\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 31, normalized size = 0.86 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {erf}\left (b x + a\right ) + \frac {e^{\left (-{\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 47, normalized size = 1.31 \begin {gather*} \frac {{\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.31, size = 53, normalized size = 1.47 \begin {gather*} \begin {cases} \frac {a \operatorname {erf}{\left (a + b x \right )}}{b} + x \operatorname {erf}{\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 {erf}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 59, normalized size = 1.64 \begin {gather*} x \operatorname {erf}\left (b x + a\right ) - \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 }} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.22, size = 48, normalized size = 1.33 \begin {gather*} x\,\mathrm {erf}\left (a+b\,x\right )+\frac {a\,\mathrm {erf}\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 }} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________