Optimal. Leaf size=53 \[ \frac {e^{c+d x^2} \text {Erfi}(b x)}{2 d}-\frac {b e^c \text {Erfi}\left (\sqrt {b^2+d} x\right )}{2 d \sqrt {b^2+d}} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, 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 = {6519, 2235}
\begin {gather*} \frac {\text {Erfi}(b x) e^{c+d x^2}}{2 d}-\frac {b e^c \text {Erfi}\left (x \sqrt {b^2+d}\right )}{2 d \sqrt {b^2+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 6519
Rubi steps
\begin {align*} \int e^{c+d x^2} x \text {erfi}(b x) \, dx &=\frac {e^{c+d x^2} \text {erfi}(b x)}{2 d}-\frac {b \int e^{c+\left (b^2+d\right ) x^2} \, dx}{d \sqrt {\pi }}\\ &=\frac {e^{c+d x^2} \text {erfi}(b x)}{2 d}-\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{2 d \sqrt {b^2+d}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 0.89 \begin {gather*} \frac {e^c \left (e^{d x^2} \text {Erfi}(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.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{d \,x^{2}+c} x \erfi \left (b x \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 61, normalized size = 1.15 \begin {gather*} \frac {\sqrt {-b^{2} - d} b \operatorname {erf}\left (\sqrt {-b^{2} - d} x\right ) e^{c} + {\left (b^{2} + d\right )} \operatorname {erfi}\left (b x\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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int x e^{d x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 51, normalized size = 0.96 \begin {gather*} \frac {{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (b\,x\right )}{2\,d}-\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2-d}\right )}{2\,d\,\sqrt {-b^2-d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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