Optimal. Leaf size=71 \[ -\frac{x e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } b}+\frac{\text{Erfi}(b x)^2}{4 b^2}+\frac{e^{2 b^2 x^2}}{2 \pi b^2}+\frac{1}{2} x^2 \text{Erfi}(b x)^2 \]
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Rubi [A] time = 0.0766503, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {6366, 6387, 6375, 30, 2209} \[ -\frac{x e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } b}+\frac{\text{Erfi}(b x)^2}{4 b^2}+\frac{e^{2 b^2 x^2}}{2 \pi b^2}+\frac{1}{2} x^2 \text{Erfi}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6366
Rule 6387
Rule 6375
Rule 30
Rule 2209
Rubi steps
\begin{align*} \int x \text{erfi}(b x)^2 \, dx &=\frac{1}{2} x^2 \text{erfi}(b x)^2-\frac{(2 b) \int e^{b^2 x^2} x^2 \text{erfi}(b x) \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{b^2 x^2} x \text{erfi}(b x)}{b \sqrt{\pi }}+\frac{1}{2} x^2 \text{erfi}(b x)^2+\frac{2 \int e^{2 b^2 x^2} x \, dx}{\pi }+\frac{\int e^{b^2 x^2} \text{erfi}(b x) \, dx}{b \sqrt{\pi }}\\ &=\frac{e^{2 b^2 x^2}}{2 b^2 \pi }-\frac{e^{b^2 x^2} x \text{erfi}(b x)}{b \sqrt{\pi }}+\frac{1}{2} x^2 \text{erfi}(b x)^2+\frac{\operatorname{Subst}(\int x \, dx,x,\text{erfi}(b x))}{2 b^2}\\ &=\frac{e^{2 b^2 x^2}}{2 b^2 \pi }-\frac{e^{b^2 x^2} x \text{erfi}(b x)}{b \sqrt{\pi }}+\frac{\text{erfi}(b x)^2}{4 b^2}+\frac{1}{2} x^2 \text{erfi}(b x)^2\\ \end{align*}
Mathematica [A] time = 0.0160232, size = 63, normalized size = 0.89 \[ \frac{\left (2 \pi b^2 x^2+\pi \right ) \text{Erfi}(b x)^2-4 \sqrt{\pi } b x e^{b^2 x^2} \text{Erfi}(b x)+2 e^{2 b^2 x^2}}{4 \pi b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int x \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 x \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 = 2.30628, size = 143, normalized size = 2.01 \begin{align*} -\frac{4 \, \sqrt{\pi } b x \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} -{\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname{erfi}\left (b x\right )^{2} - 2 \, e^{\left (2 \, b^{2} x^{2}\right )}}{4 \, \pi b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.52173, size = 63, normalized size = 0.89 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{erfi}^{2}{\left (b x \right )}}{2} - \frac{x e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{\sqrt{\pi } b} + \frac{e^{2 b^{2} x^{2}}}{2 \pi b^{2}} + \frac{\operatorname{erfi}^{2}{\left (b x \right )}}{4 b^{2}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \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 x \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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