Optimal. Leaf size=124 \[ -\frac{x^3 e^{b^2 x^2} \text{Erfi}(b x)}{2 \sqrt{\pi } b}+\frac{3 x e^{b^2 x^2} \text{Erfi}(b x)}{4 \sqrt{\pi } b^3}-\frac{3 \text{Erfi}(b x)^2}{16 b^4}+\frac{x^2 e^{2 b^2 x^2}}{4 \pi b^2}-\frac{e^{2 b^2 x^2}}{2 \pi b^4}+\frac{1}{4} x^4 \text{Erfi}(b x)^2 \]
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Rubi [A] time = 0.159084, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6366, 6387, 6375, 30, 2209, 2212} \[ -\frac{x^3 e^{b^2 x^2} \text{Erfi}(b x)}{2 \sqrt{\pi } b}+\frac{3 x e^{b^2 x^2} \text{Erfi}(b x)}{4 \sqrt{\pi } b^3}-\frac{3 \text{Erfi}(b x)^2}{16 b^4}+\frac{x^2 e^{2 b^2 x^2}}{4 \pi b^2}-\frac{e^{2 b^2 x^2}}{2 \pi b^4}+\frac{1}{4} x^4 \text{Erfi}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 6366
Rule 6387
Rule 6375
Rule 30
Rule 2209
Rule 2212
Rubi steps
\begin{align*} \int x^3 \text{erfi}(b x)^2 \, dx &=\frac{1}{4} x^4 \text{erfi}(b x)^2-\frac{b \int e^{b^2 x^2} x^4 \text{erfi}(b x) \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{b^2 x^2} x^3 \text{erfi}(b x)}{2 b \sqrt{\pi }}+\frac{1}{4} x^4 \text{erfi}(b x)^2+\frac{\int e^{2 b^2 x^2} x^3 \, dx}{\pi }+\frac{3 \int e^{b^2 x^2} x^2 \text{erfi}(b x) \, dx}{2 b \sqrt{\pi }}\\ &=\frac{e^{2 b^2 x^2} x^2}{4 b^2 \pi }+\frac{3 e^{b^2 x^2} x \text{erfi}(b x)}{4 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^3 \text{erfi}(b x)}{2 b \sqrt{\pi }}+\frac{1}{4} x^4 \text{erfi}(b x)^2-\frac{\int e^{2 b^2 x^2} x \, dx}{2 b^2 \pi }-\frac{3 \int e^{2 b^2 x^2} x \, dx}{2 b^2 \pi }-\frac{3 \int e^{b^2 x^2} \text{erfi}(b x) \, dx}{4 b^3 \sqrt{\pi }}\\ &=-\frac{e^{2 b^2 x^2}}{2 b^4 \pi }+\frac{e^{2 b^2 x^2} x^2}{4 b^2 \pi }+\frac{3 e^{b^2 x^2} x \text{erfi}(b x)}{4 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^3 \text{erfi}(b x)}{2 b \sqrt{\pi }}+\frac{1}{4} x^4 \text{erfi}(b x)^2-\frac{3 \operatorname{Subst}(\int x \, dx,x,\text{erfi}(b x))}{8 b^4}\\ &=-\frac{e^{2 b^2 x^2}}{2 b^4 \pi }+\frac{e^{2 b^2 x^2} x^2}{4 b^2 \pi }+\frac{3 e^{b^2 x^2} x \text{erfi}(b x)}{4 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^3 \text{erfi}(b x)}{2 b \sqrt{\pi }}-\frac{3 \text{erfi}(b x)^2}{16 b^4}+\frac{1}{4} x^4 \text{erfi}(b x)^2\\ \end{align*}
Mathematica [A] time = 0.027615, size = 82, normalized size = 0.66 \[ \frac{\pi \left (4 b^4 x^4-3\right ) \text{Erfi}(b x)^2-4 \sqrt{\pi } b x e^{b^2 x^2} \left (2 b^2 x^2-3\right ) \text{Erfi}(b x)+4 e^{2 b^2 x^2} \left (b^2 x^2-2\right )}{16 \pi b^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \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^{3} \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 = 2.41216, size = 188, normalized size = 1.52 \begin{align*} -\frac{4 \, \sqrt{\pi }{\left (2 \, b^{3} x^{3} - 3 \, b x\right )} \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} +{\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname{erfi}\left (b x\right )^{2} - 4 \,{\left (b^{2} x^{2} - 2\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.10517, size = 116, normalized size = 0.94 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{erfi}^{2}{\left (b x \right )}}{4} - \frac{x^{3} e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{2 \sqrt{\pi } b} + \frac{x^{2} e^{2 b^{2} x^{2}}}{4 \pi b^{2}} + \frac{3 x e^{b^{2} x^{2}} \operatorname{erfi}{\left (b x \right )}}{4 \sqrt{\pi } b^{3}} - \frac{e^{2 b^{2} x^{2}}}{2 \pi b^{4}} - \frac{3 \operatorname{erfi}^{2}{\left (b x \right )}}{16 b^{4}} & \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^{3} \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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