Optimal. Leaf size=27 \[ -\frac {b \text {Ei}\left (-b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6362, 2210} \[ -\frac {b \text {Ei}\left (-b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {Erfc}(b x)}{x} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 6362
Rubi steps
\begin {align*} \int \frac {\text {erfc}(b x)}{x^2} \, dx &=-\frac {\text {erfc}(b x)}{x}-\frac {(2 b) \int \frac {e^{-b^2 x^2}}{x} \, dx}{\sqrt {\pi }}\\ &=-\frac {\text {erfc}(b x)}{x}-\frac {b \text {Ei}\left (-b^2 x^2\right )}{\sqrt {\pi }}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ -\frac {b \text {Ei}\left (-b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 32, normalized size = 1.19 \[ -\frac {\pi + \sqrt {\pi } b x {\rm Ei}\left (-b^{2} x^{2}\right ) - \pi \operatorname {erf}\left (b x\right )}{\pi x} \]
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 \[ \int \frac {\operatorname {erfc}\left (b x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.07 \[ b \left (-\frac {\mathrm {erfc}\left (b x \right )}{b x}+\frac {\Ei \left (1, b^{2} x^{2}\right )}{\sqrt {\pi }}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 25, normalized size = 0.93 \[ -\frac {b {\rm Ei}\left (-b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erfc}\left (b x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 25, normalized size = 0.93 \[ -\frac {\mathrm {erfc}\left (b\,x\right )}{x}-\frac {b\,\mathrm {ei}\left (-b^2\,x^2\right )}{\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.98, size = 20, normalized size = 0.74 \[ \frac {b \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erfc}{\left (b x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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