Optimal. Leaf size=25 \[ \frac{b \text{ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfi}(b x)}{x} \]
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Rubi [A] time = 0.0291507, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6363, 2210} \[ \frac{b \text{Ei}\left (b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfi}(b x)}{x} \]
Antiderivative was successfully verified.
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Rule 6363
Rule 2210
Rubi steps
\begin{align*} \int \frac{\text{erfi}(b x)}{x^2} \, dx &=-\frac{\text{erfi}(b x)}{x}+\frac{(2 b) \int \frac{e^{b^2 x^2}}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{\text{erfi}(b x)}{x}+\frac{b \text{Ei}\left (b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.0122441, size = 25, normalized size = 1. \[ \frac{b \text{ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfi}(b x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 31, normalized size = 1.2 \begin{align*} b \left ( -{\frac{{\it erfi} \left ( bx \right ) }{bx}}-{\frac{{\it Ei} \left ( 1,-{b}^{2}{x}^{2} \right ) }{\sqrt{\pi }}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.29618, size = 31, normalized size = 1.24 \begin{align*} \frac{b{\rm Ei}\left (b^{2} x^{2}\right )}{\sqrt{\pi }} - \frac{\operatorname{erfi}\left (b x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35493, size = 68, normalized size = 2.72 \begin{align*} \frac{\sqrt{\pi } b x{\rm Ei}\left (b^{2} x^{2}\right ) - \pi \operatorname{erfi}\left (b x\right )}{\pi x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.2032, size = 32, normalized size = 1.28 \begin{align*} - \frac{b \operatorname{E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{\sqrt{\pi }} - \frac{i \operatorname{erfc}{\left (i b x \right )}}{x} + \frac{i}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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