Optimal. Leaf size=59 \[ -\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{x}+\frac{b e^c \text{ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt{\pi }}+\frac{1}{2} \sqrt{\pi } b e^c \text{Erfi}(b x)^2 \]
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Rubi [A] time = 0.0816846, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6393, 6375, 30, 2210} \[ -\frac{e^{b^2 x^2+c} \text{Erfi}(b x)}{x}+\frac{b e^c \text{Ei}\left (2 b^2 x^2\right )}{\sqrt{\pi }}+\frac{1}{2} \sqrt{\pi } b e^c \text{Erfi}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6393
Rule 6375
Rule 30
Rule 2210
Rubi steps
\begin{align*} \int \frac{e^{c+b^2 x^2} \text{erfi}(b x)}{x^2} \, dx &=-\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{x}+\left (2 b^2\right ) \int e^{c+b^2 x^2} \text{erfi}(b x) \, dx+\frac{(2 b) \int \frac{e^{c+2 b^2 x^2}}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{x}+\frac{b e^c \text{Ei}\left (2 b^2 x^2\right )}{\sqrt{\pi }}+\left (b e^c \sqrt{\pi }\right ) \operatorname{Subst}(\int x \, dx,x,\text{erfi}(b x))\\ &=-\frac{e^{c+b^2 x^2} \text{erfi}(b x)}{x}+\frac{1}{2} b e^c \sqrt{\pi } \text{erfi}(b x)^2+\frac{b e^c \text{Ei}\left (2 b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.0210227, size = 56, normalized size = 0.95 \[ \frac{1}{2} e^c \left (-\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{x}+\frac{2 b \text{ExpIntegralEi}\left (2 b^2 x^2\right )}{\sqrt{\pi }}+\sqrt{\pi } b \text{Erfi}(b x)^2\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfi} \left ( bx \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69375, size = 135, normalized size = 2.29 \begin{align*} -\frac{{\left (2 \, \pi \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - \sqrt{\pi }{\left (\pi b x \operatorname{erfi}\left (b x\right )^{2} + 2 \, b x{\rm Ei}\left (2 \, b^{2} x^{2}\right )\right )}\right )} e^{c}}{2 \, \pi x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int \frac{e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{x^{2}}\, dx \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 ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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