Optimal. Leaf size=60 \[ -\frac {2 b^3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x}+\frac {2 b \log (x)}{\sqrt {\pi }} \]
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Rubi [A] time = 0.06, antiderivative size = 60, 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, 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 29
Rule 6378
Rule 6393
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*}
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Mathematica [C] time = 0.02, 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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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\erfi \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{2}}\, dx \]
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 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{x^2} \,d x \]
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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