Optimal. Leaf size=57 \[ \frac {b e^c \text {Erf}\left (\sqrt {b^2-d} x\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {Erfc}(b x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6518, 2236}
\begin {gather*} \frac {b e^c \text {Erf}\left (x \sqrt {b^2-d}\right )}{2 d \sqrt {b^2-d}}+\frac {\text {Erfc}(b x) e^{c+d x^2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2236
Rule 6518
Rubi steps
\begin {align*} \int e^{c+d x^2} x \text {erfc}(b x) \, dx &=\frac {e^{c+d x^2} \text {erfc}(b x)}{2 d}+\frac {b \int e^{c-\left (b^2-d\right ) x^2} \, dx}{d \sqrt {\pi }}\\ &=\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {erfc}(b x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 50, normalized size = 0.88 \begin {gather*} \frac {e^c \left (e^{d x^2} \text {Erfc}(b x)+\frac {b \text {Erfi}\left (\sqrt {-b^2+d} x\right )}{\sqrt {-b^2+d}}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.44, size = 92, normalized size = 1.61
method | result | size |
default | \(\frac {\frac {b \,{\mathrm e}^{\frac {b^{2} d \,x^{2}+b^{2} c}{b^{2}}}}{2 d}-\frac {\erf \left (b x \right ) b \,{\mathrm e}^{\frac {b^{2} d \,x^{2}+b^{2} c}{b^{2}}}}{2 d}+\frac {b \,{\mathrm e}^{c} \erf \left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{2 d \sqrt {1-\frac {d}{b^{2}}}}}{b}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 70, normalized size = 1.23 \begin {gather*} \frac {\sqrt {b^{2} - d} b \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c} + {\left (b^{2} - {\left (b^{2} - d\right )} \operatorname {erf}\left (b x\right ) - d\right )} e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d - d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int x e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________