Optimal. Leaf size=65 \[ -\frac {2 b^3 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-\frac {b}{\sqrt {\pi } x} \]
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Rubi [A] time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 30
Rule 6390
Rule 6393
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*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.49 \[ -\frac {2 b \, _2F_2\left (-\frac {1}{2},1;\frac {1}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi } x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\erfi \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{3}}\, dx \]
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 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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