Optimal. Leaf size=65 \[ -\frac{2 b^3 x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 x^2}-\frac{b}{\sqrt{\pi } x} \]
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Rubi [A] time = 0.0748705, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6393, 6390, 30} \[ -\frac{2 b^3 x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};-b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 x^2}-\frac{b}{\sqrt{\pi } x} \]
Antiderivative was successfully verified.
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Rule 6393
Rule 6390
Rule 30
Rubi steps
\begin{align*} \int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x^3} \, dx &=-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{2 x^2}-b^2 \int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x} \, dx+\frac{b \int \frac{1}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{b}{\sqrt{\pi } x}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{2 x^2}-\frac{2 b^3 x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};-b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.0146599, size = 32, normalized size = 0.49 \[ -\frac{2 b \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},1\right \},\left \{\frac{1}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi } x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.189, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfi} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{3}}}\, 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 \frac{\operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{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 (\frac{\operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}, 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 \frac{\operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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