Optimal. Leaf size=21 \[ \frac {\sqrt {\pi } e^c \text {erfi}(b x)^2}{4 b} \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6375, 30} \[ \frac {\sqrt {\pi } e^c \text {Erfi}(b x)^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6375
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} \text {erfi}(b x) \, dx &=\frac {\left (e^c \sqrt {\pi }\right ) \operatorname {Subst}(\int x \, dx,x,\text {erfi}(b x))}{2 b}\\ &=\frac {e^c \sqrt {\pi } \text {erfi}(b x)^2}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \frac {\sqrt {\pi } e^c \text {erfi}(b x)^2}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 16, normalized size = 0.76 \[ \frac {\sqrt {\pi } \operatorname {erfi}\left (b x\right )^{2} e^{c}}{4 \, b} \]
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 \[ \int \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b^{2} x^{2}+c} \erfi \left (b x \right )\, dx \]
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 \[ \int \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 91, normalized size = 4.33 \[ \frac {\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )\,{\mathrm {e}}^c\,\mathrm {erfi}\left (b\,x\right )}{2\,\sqrt {b^2}}-\frac {\sqrt {\pi }\,{\mathrm {e}}^c\,{\mathrm {erf}\left (x\,\sqrt {-b^2}\right )}^2}{4\,b}-\frac {b\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b^2\,x}{\sqrt {b^2}}\right )\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2}\right )}{2\,\sqrt {b^2}\,\sqrt {-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 19, normalized size = 0.90 \[ \begin {cases} \frac {\sqrt {\pi } e^{c} \operatorname {erfi}^{2}{\left (b x \right )}}{4 b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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