Optimal. Leaf size=54 \[ -\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } b}+x \text{Erfi}(b x)^2+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} b x\right )}{b} \]
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Rubi [A] time = 0.0445048, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6354, 12, 6384, 2204} \[ -\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } b}+x \text{Erfi}(b x)^2+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} b x\right )}{b} \]
Antiderivative was successfully verified.
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Rule 6354
Rule 12
Rule 6384
Rule 2204
Rubi steps
\begin{align*} \int \text{erfi}(b x)^2 \, dx &=x \text{erfi}(b x)^2-\frac{4 \int b e^{b^2 x^2} x \text{erfi}(b x) \, dx}{\sqrt{\pi }}\\ &=x \text{erfi}(b x)^2-\frac{(4 b) \int e^{b^2 x^2} x \text{erfi}(b x) \, dx}{\sqrt{\pi }}\\ &=-\frac{2 e^{b^2 x^2} \text{erfi}(b x)}{b \sqrt{\pi }}+x \text{erfi}(b x)^2+\frac{4 \int e^{2 b^2 x^2} \, dx}{\pi }\\ &=-\frac{2 e^{b^2 x^2} \text{erfi}(b x)}{b \sqrt{\pi }}+x \text{erfi}(b x)^2+\frac{\sqrt{\frac{2}{\pi }} \text{erfi}\left (\sqrt{2} b x\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.011072, size = 54, normalized size = 1. \[ -\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } b}+x \text{Erfi}(b x)^2+\frac{\sqrt{\frac{2}{\pi }} \text{Erfi}\left (\sqrt{2} b x\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ({\it erfi} \left ( bx \right ) \right ) ^{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 \operatorname{erfi}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43849, size = 169, normalized size = 3.13 \begin{align*} \frac{\pi b^{2} x \operatorname{erfi}\left (b x\right )^{2} - 2 \, \sqrt{\pi } b \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + \sqrt{2} \sqrt{\pi } \sqrt{b^{2}} \operatorname{erfi}\left (\sqrt{2} \sqrt{b^{2}} x\right )}{\pi b^{2}} \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*} \int \operatorname{erfi}^{2}{\left (b x \right )}\, 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 \operatorname{erfi}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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