Optimal. Leaf size=86 \[ \frac {e^{c+d x^2} \text {erf}(a+b x)}{2 d}-\frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \sqrt {b^2-d}} \]
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Rubi [A] time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6382, 2234, 2205} \[ \frac {e^{c+d x^2} \text {Erf}(a+b x)}{2 d}-\frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {Erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \sqrt {b^2-d}} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2234
Rule 6382
Rubi steps
\begin {align*} \int e^{c+d x^2} x \text {erf}(a+b x) \, dx &=\frac {e^{c+d x^2} \text {erf}(a+b x)}{2 d}-\frac {b \int e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} \, dx}{d \sqrt {\pi }}\\ &=\frac {e^{c+d x^2} \text {erf}(a+b x)}{2 d}-\frac {\left (b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{d \sqrt {\pi }}\\ &=\frac {e^{c+d x^2} \text {erf}(a+b x)}{2 d}-\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 82, normalized size = 0.95 \[ \frac {e^c \left (e^{d x^2} \text {erf}(a+b x)-\frac {b e^{\frac {a^2 d}{b^2-d}} \text {erfi}\left (\frac {x \left (d-b^2\right )-a b}{\sqrt {d-b^2}}\right )}{\sqrt {d-b^2}}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 100, normalized size = 1.16 \[ -\frac {\sqrt {b^{2} - d} b \operatorname {erf}\left (\frac {a b + {\left (b^{2} - d\right )} x}{\sqrt {b^{2} - d}}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} - c\right )} d}{b^{2} - d}\right )} - {\left (b^{2} - d\right )} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d - d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 87, normalized size = 1.01 \[ \frac {b \operatorname {erf}\left (-\sqrt {b^{2} - d} {\left (\frac {a b}{b^{2} - d} + x\right )}\right ) e^{\left (\frac {b^{2} c + a^{2} d - c d}{b^{2} - d}\right )}}{2 \, \sqrt {b^{2} - d} d} + \frac {\operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 134, normalized size = 1.56 \[ \frac {\frac {\erf \left (b x +a \right ) b \,{\mathrm e}^{\frac {\left (b x +a \right )^{2} d}{b^{2}}-\frac {2 a d \left (b x +a \right )}{b^{2}}+\frac {a^{2} d}{b^{2}}+c}}{2 d}-\frac {b \,{\mathrm e}^{\frac {a^{2} d}{b^{2}}+c -\frac {a^{2} d^{2}}{b^{4} \left (-1+\frac {d}{b^{2}}\right )}} \erf \left (\sqrt {1-\frac {d}{b^{2}}}\, \left (b x +a \right )+\frac {a d}{b^{2} \sqrt {1-\frac {d}{b^{2}}}}\right )}{2 d \sqrt {1-\frac {d}{b^{2}}}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 84, normalized size = 0.98 \[ -\frac {b \operatorname {erf}\left (\frac {a b}{\sqrt {b^{2} - d}} + \sqrt {b^{2} - d} x\right ) e^{\left (\frac {a^{2} b^{2}}{b^{2} - d} - a^{2} + c\right )}}{2 \, \sqrt {b^{2} - d} d} + \frac {\operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 89, normalized size = 1.03 \[ \frac {\mathrm {erf}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}-\frac {b\,\mathrm {erf}\left (\frac {a\,b\,1{}\mathrm {i}-x\,\left (d-b^2\right )\,1{}\mathrm {i}}{\sqrt {d-b^2}}\right )\,{\mathrm {e}}^{c-a^2-\frac {a^2\,b^2}{d-b^2}}\,1{}\mathrm {i}}{2\,d\,\sqrt {d-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x e^{d x^{2}} \operatorname {erf}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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