Optimal. Leaf size=84 \[ \frac{x^4 e^{-b^2 x^2}}{5 \sqrt{\pi } b}+\frac{2 x^2 e^{-b^2 x^2}}{5 \sqrt{\pi } b^3}+\frac{2 e^{-b^2 x^2}}{5 \sqrt{\pi } b^5}+\frac{1}{5} x^5 \text{Erf}(b x) \]
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Rubi [A] time = 0.0733966, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6361, 2212, 2209} \[ \frac{x^4 e^{-b^2 x^2}}{5 \sqrt{\pi } b}+\frac{2 x^2 e^{-b^2 x^2}}{5 \sqrt{\pi } b^3}+\frac{2 e^{-b^2 x^2}}{5 \sqrt{\pi } b^5}+\frac{1}{5} x^5 \text{Erf}(b x) \]
Antiderivative was successfully verified.
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Rule 6361
Rule 2212
Rule 2209
Rubi steps
\begin{align*} \int x^4 \text{erf}(b x) \, dx &=\frac{1}{5} x^5 \text{erf}(b x)-\frac{(2 b) \int e^{-b^2 x^2} x^5 \, dx}{5 \sqrt{\pi }}\\ &=\frac{e^{-b^2 x^2} x^4}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erf}(b x)-\frac{4 \int e^{-b^2 x^2} x^3 \, dx}{5 b \sqrt{\pi }}\\ &=\frac{2 e^{-b^2 x^2} x^2}{5 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^4}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erf}(b x)-\frac{4 \int e^{-b^2 x^2} x \, dx}{5 b^3 \sqrt{\pi }}\\ &=\frac{2 e^{-b^2 x^2}}{5 b^5 \sqrt{\pi }}+\frac{2 e^{-b^2 x^2} x^2}{5 b^3 \sqrt{\pi }}+\frac{e^{-b^2 x^2} x^4}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erf}(b x)\\ \end{align*}
Mathematica [A] time = 0.0161336, size = 66, normalized size = 0.79 \[ e^{-b^2 x^2} \left (\frac{2 x^2}{5 \sqrt{\pi } b^3}+\frac{2}{5 \sqrt{\pi } b^5}+\frac{x^4}{5 \sqrt{\pi } b}\right )+\frac{1}{5} x^5 \text{Erf}(b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 72, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{5}} \left ({\frac{{\it Erf} \left ( bx \right ){b}^{5}{x}^{5}}{5}}-{\frac{2}{5\,\sqrt{\pi }} \left ( -{\frac{{b}^{4}{x}^{4}}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}}}-{\frac{{b}^{2}{x}^{2}}{{{\rm e}^{{b}^{2}{x}^{2}}}}}- \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{-1} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04654, size = 59, normalized size = 0.7 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{erf}\left (b x\right ) + \frac{{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt{\pi } b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52465, size = 116, normalized size = 1.38 \begin{align*} \frac{\pi b^{5} x^{5} \operatorname{erf}\left (b x\right ) + \sqrt{\pi }{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \pi b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.46533, size = 75, normalized size = 0.89 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{erf}{\left (b x \right )}}{5} + \frac{x^{4} e^{- b^{2} x^{2}}}{5 \sqrt{\pi } b} + \frac{2 x^{2} e^{- b^{2} x^{2}}}{5 \sqrt{\pi } b^{3}} + \frac{2 e^{- b^{2} x^{2}}}{5 \sqrt{\pi } b^{5}} & \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 [A] time = 1.2741, size = 59, normalized size = 0.7 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{erf}\left (b x\right ) + \frac{{\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} e^{\left (-b^{2} x^{2}\right )}}{5 \, \sqrt{\pi } b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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