Optimal. Leaf size=42 \[ b^2 (-\text {erf}(b x))-\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erf}(b x)}{2 x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 42, 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 = {6361, 2214, 2205} \[ b^2 (-\text {Erf}(b x))-\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {Erf}(b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2214
Rule 6361
Rubi steps
\begin {align*} \int \frac {\text {erf}(b x)}{x^3} \, dx &=-\frac {\text {erf}(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 {erf}(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 {erf}(b x)-\frac {\text {erf}(b x)}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 42, normalized size = 1.00 \[ b^2 (-\text {erf}(b x))-\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erf}(b x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 41, normalized size = 0.98 \[ -\frac {2 \, \sqrt {\pi } b x e^{\left (-b^{2} x^{2}\right )} + {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )}{2 \, \pi x^{2}} \]
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 \[ \int \frac {\operatorname {erf}\left (b x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 1.19 \[ b^{2} \left (-\frac {\erf \left (b x \right )}{2 b^{2} x^{2}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{b x}-\sqrt {\pi }\, \erf \left (b x \right )}{\sqrt {\pi }}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 35, normalized size = 0.83 \[ -\frac {b^{2} \sqrt {x^{2}} \Gamma \left (-\frac {1}{2}, b^{2} x^{2}\right )}{2 \, \sqrt {\pi } x} - \frac {\operatorname {erf}\left (b x\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 67, normalized size = 1.60 \[ \frac {b\,\mathrm {erfc}\left (\sqrt {b^2\,x^2}\right )\,\sqrt {b^2\,x^2}}{x}-\frac {b\,\sqrt {b^2\,x^2}}{x}-\frac {b\,{\mathrm {e}}^{-b^2\,x^2}}{x\,\sqrt {\pi }}-\frac {\mathrm {erf}\left (b\,x\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 36, normalized size = 0.86 \[ - b^{2} \operatorname {erf}{\left (b x \right )} - \frac {b e^{- b^{2} x^{2}}}{\sqrt {\pi } x} - \frac {\operatorname {erf}{\left (b x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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