Optimal. Leaf size=27 \[ \frac{b x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{\sqrt{\pi }} \]
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Rubi [A] time = 0.0160768, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {6378} \[ \frac{b x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{\sqrt{\pi }} \]
Antiderivative was successfully verified.
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Rule 6378
Rubi steps
\begin{align*} \int e^{-b^2 x^2} \text{erfi}(b x) \, dx &=\frac{b x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.0084946, size = 27, normalized size = 1. \[ \frac{b x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{\sqrt{\pi }} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfi} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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