Optimal. Leaf size=40 \[ b^2 \text {erfi}(b x)-\frac {b e^{b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erfi}(b x)}{2 x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6363, 2214, 2204} \[ b^2 \text {Erfi}(b x)-\frac {b e^{b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {Erfi}(b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2214
Rule 6363
Rubi steps
\begin {align*} \int \frac {\text {erfi}(b x)}{x^3} \, dx &=-\frac {\text {erfi}(b x)}{2 x^2}+\frac {b \int \frac {e^{b^2 x^2}}{x^2} \, dx}{\sqrt {\pi }}\\ &=-\frac {b e^{b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erfi}(b x)}{2 x^2}+\frac {\left (2 b^3\right ) \int e^{b^2 x^2} \, dx}{\sqrt {\pi }}\\ &=-\frac {b e^{b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {erfi}(b x)-\frac {\text {erfi}(b x)}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 0.92 \[ \left (b^2-\frac {1}{2 x^2}\right ) \text {erfi}(b x)-\frac {b e^{b^2 x^2}}{\sqrt {\pi } x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 40, normalized size = 1.00 \[ -\frac {2 \, \sqrt {\pi } b x e^{\left (b^{2} x^{2}\right )} + {\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )}{2 \, \pi x^{2}} \]
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 \[ \int \frac {\operatorname {erfi}\left (b x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 1.18 \[ b^{2} \left (-\frac {\erfi \left (b x \right )}{2 b^{2} x^{2}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{b x}+\sqrt {\pi }\, \erfi \left (b x \right )}{\sqrt {\pi }}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 39, normalized size = 0.98 \[ -\frac {\sqrt {-b^{2} x^{2}} b \Gamma \left (-\frac {1}{2}, -b^{2} x^{2}\right )}{2 \, \sqrt {\pi } x} - \frac {\operatorname {erfi}\left (b x\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 69, normalized size = 1.72 \[ \frac {b\,\mathrm {erfc}\left (\sqrt {-b^2\,x^2}\right )\,\sqrt {-b^2\,x^2}}{x}-\frac {b\,\sqrt {-b^2\,x^2}}{x}-\frac {b\,{\mathrm {e}}^{b^2\,x^2}}{x\,\sqrt {\pi }}-\frac {\mathrm {erfi}\left (b\,x\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 34, normalized size = 0.85 \[ b^{2} \operatorname {erfi}{\left (b x \right )} - \frac {b e^{b^{2} x^{2}}}{\sqrt {\pi } x} - \frac {\operatorname {erfi}{\left (b x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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