Optimal. Leaf size=107 \[ -\frac{x^4 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{x^2 e^{-b^2 x^2} \text{Erfi}(b x)}{b^4}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{b^6}+\frac{2 x^3}{3 \sqrt{\pi } b^3}+\frac{2 x}{\sqrt{\pi } b^5}+\frac{x^5}{5 \sqrt{\pi } b} \]
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Rubi [A] time = 0.115105, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6387, 6384, 8, 30} \[ -\frac{x^4 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{x^2 e^{-b^2 x^2} \text{Erfi}(b x)}{b^4}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{b^6}+\frac{2 x^3}{3 \sqrt{\pi } b^3}+\frac{2 x}{\sqrt{\pi } b^5}+\frac{x^5}{5 \sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Rule 6387
Rule 6384
Rule 8
Rule 30
Rubi steps
\begin{align*} \int e^{-b^2 x^2} x^5 \text{erfi}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^4 \text{erfi}(b x)}{2 b^2}+\frac{2 \int e^{-b^2 x^2} x^3 \text{erfi}(b x) \, dx}{b^2}+\frac{\int x^4 \, dx}{b \sqrt{\pi }}\\ &=\frac{x^5}{5 b \sqrt{\pi }}-\frac{e^{-b^2 x^2} x^2 \text{erfi}(b x)}{b^4}-\frac{e^{-b^2 x^2} x^4 \text{erfi}(b x)}{2 b^2}+\frac{2 \int e^{-b^2 x^2} x \text{erfi}(b x) \, dx}{b^4}+\frac{2 \int x^2 \, dx}{b^3 \sqrt{\pi }}\\ &=\frac{2 x^3}{3 b^3 \sqrt{\pi }}+\frac{x^5}{5 b \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{b^6}-\frac{e^{-b^2 x^2} x^2 \text{erfi}(b x)}{b^4}-\frac{e^{-b^2 x^2} x^4 \text{erfi}(b x)}{2 b^2}+\frac{2 \int 1 \, dx}{b^5 \sqrt{\pi }}\\ &=\frac{2 x}{b^5 \sqrt{\pi }}+\frac{2 x^3}{3 b^3 \sqrt{\pi }}+\frac{x^5}{5 b \sqrt{\pi }}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{b^6}-\frac{e^{-b^2 x^2} x^2 \text{erfi}(b x)}{b^4}-\frac{e^{-b^2 x^2} x^4 \text{erfi}(b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0436194, size = 68, normalized size = 0.64 \[ \frac{\frac{6 b^5 x^5+20 b^3 x^3+60 b x}{\sqrt{\pi }}-15 e^{-b^2 x^2} \left (b^4 x^4+2 b^2 x^2+2\right ) \text{Erfi}(b x)}{30 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.303, size = 103, normalized size = 1. \begin{align*}{\frac{6\,{x}^{5}{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{5}-15\,{\it erfi} \left ( bx \right ){x}^{4}{b}^{4}\sqrt{\pi }+20\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{3}{x}^{3}-30\,\sqrt{\pi }{\it erfi} \left ( bx \right ){b}^{2}{x}^{2}+60\,{{\rm e}^{{b}^{2}{x}^{2}}}bx-30\,\sqrt{\pi }{\it erfi} \left ( bx \right ) }{30\,{b}^{6}\sqrt{\pi }{{\rm e}^{{b}^{2}{x}^{2}}}}} \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^{5} \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58599, size = 185, normalized size = 1.73 \begin{align*} \frac{{\left (2 \, \sqrt{\pi }{\left (3 \, b^{5} x^{5} + 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - 15 \,{\left (2 \, \pi + \pi b^{4} x^{4} + 2 \, \pi b^{2} x^{2}\right )} \operatorname{erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{30 \, \pi b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{5} \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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