Optimal. Leaf size=47 \[ \frac {e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^c \text {erfi}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6384, 2204} \[ \frac {e^{b^2 x^2+c} \text {Erfi}(b x)}{2 b^2}-\frac {e^c \text {Erfi}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 6384
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx &=\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\int e^{c+2 b^2 x^2} \, dx}{b \sqrt {\pi }}\\ &=\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^c \text {erfi}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.89 \[ \frac {e^c \left (2 e^{b^2 x^2} \text {erfi}(b x)-\sqrt {2} \text {erfi}\left (\sqrt {2} b x\right )\right )}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 46, normalized size = 0.98 \[ \frac {2 \, b \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} - \sqrt {2} \sqrt {b^{2}} \operatorname {erfi}\left (\sqrt {2} \sqrt {b^{2}} x\right ) e^{c}}{4 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b^{2} x^{2}+c} x \erfi \left (b x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 50, normalized size = 1.06 \[ \frac {{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (b\,x\right )}{2\,b^2}-\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )\,{\mathrm {e}}^c}{4\,b\,\sqrt {-b^2}} \]
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 \[ e^{c} \int x e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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