Optimal. Leaf size=60 \[ -\frac{2 b^3 x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{x}+\frac{2 b \log (x)}{\sqrt{\pi }} \]
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Rubi [A] time = 0.0563698, antiderivative size = 60, 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, 6378, 29} \[ -\frac{2 b^3 x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{x}+\frac{2 b \log (x)}{\sqrt{\pi }} \]
Antiderivative was successfully verified.
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Rule 6393
Rule 6378
Rule 29
Rubi steps
\begin{align*} \int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x^2} \, dx &=-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{x}-\left (2 b^2\right ) \int e^{-b^2 x^2} \text{erfi}(b x) \, dx+\frac{(2 b) \int \frac{1}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{x}-\frac{2 b^3 x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{\sqrt{\pi }}+\frac{2 b \log (x)}{\sqrt{\pi }}\\ \end{align*}
Mathematica [C] time = 0.0163195, size = 26, normalized size = 0.43 \[ -\frac{1}{2} b G_{2,3}^{2,1}\left (b^2 x^2|\begin{array}{c} 0,1 \\ 0,0,-\frac{1}{2} \\\end{array}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfi} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{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 \frac{\operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}}\,{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^{2}}, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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