Optimal. Leaf size=33 \[ \frac {(-4+x) \left (2-e^{-\frac {4 x^2}{25}} x\right )}{9 x}-x \left (-2+x^2\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 43, normalized size of antiderivative = 1.30, number of steps used = 10, number of rules used = 6, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 6688, 2226, 2205, 2209, 2212} \begin {gather*} -x^3-\frac {1}{9} e^{-\frac {4 x^2}{25}} x+\frac {4}{9} e^{-\frac {4 x^2}{25}}+2 x-\frac {8}{9 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2205
Rule 2209
Rule 2212
Rule 2226
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{225} \int \frac {e^{-\frac {4 x^2}{25}} \left (-25 x^2-32 x^3+8 x^4+e^{\frac {4 x^2}{25}} \left (200+450 x^2-675 x^4\right )\right )}{x^2} \, dx\\ &=\frac {1}{225} \int \left (25 \left (18+\frac {8}{x^2}-27 x^2\right )+e^{-\frac {4 x^2}{25}} \left (-25-32 x+8 x^2\right )\right ) \, dx\\ &=\frac {1}{225} \int e^{-\frac {4 x^2}{25}} \left (-25-32 x+8 x^2\right ) \, dx+\frac {1}{9} \int \left (18+\frac {8}{x^2}-27 x^2\right ) \, dx\\ &=-\frac {8}{9 x}+2 x-x^3+\frac {1}{225} \int \left (-25 e^{-\frac {4 x^2}{25}}-32 e^{-\frac {4 x^2}{25}} x+8 e^{-\frac {4 x^2}{25}} x^2\right ) \, dx\\ &=-\frac {8}{9 x}+2 x-x^3+\frac {8}{225} \int e^{-\frac {4 x^2}{25}} x^2 \, dx-\frac {1}{9} \int e^{-\frac {4 x^2}{25}} \, dx-\frac {32}{225} \int e^{-\frac {4 x^2}{25}} x \, dx\\ &=\frac {4}{9} e^{-\frac {4 x^2}{25}}-\frac {8}{9 x}+2 x-\frac {1}{9} e^{-\frac {4 x^2}{25}} x-x^3-\frac {5}{36} \sqrt {\pi } \text {erf}\left (\frac {2 x}{5}\right )+\frac {1}{9} \int e^{-\frac {4 x^2}{25}} \, dx\\ &=\frac {4}{9} e^{-\frac {4 x^2}{25}}-\frac {8}{9 x}+2 x-\frac {1}{9} e^{-\frac {4 x^2}{25}} x-x^3\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 1.30 \begin {gather*} \frac {4}{9} e^{-\frac {4 x^2}{25}}-\frac {8}{9 x}+2 x-\frac {1}{9} e^{-\frac {4 x^2}{25}} x-x^3 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 37, normalized size = 1.12 \begin {gather*} -\frac {{\left (x^{2} + {\left (9 \, x^{4} - 18 \, x^{2} + 8\right )} e^{\left (\frac {4}{25} \, x^{2}\right )} - 4 \, x\right )} e^{\left (-\frac {4}{25} \, x^{2}\right )}}{9 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 36, normalized size = 1.09 \begin {gather*} -\frac {9 \, x^{4} + x^{2} e^{\left (-\frac {4}{25} \, x^{2}\right )} - 18 \, x^{2} - 4 \, x e^{\left (-\frac {4}{25} \, x^{2}\right )} + 8}{9 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 0.85
method | result | size |
risch | \(2 x -\frac {8}{9 x}-x^{3}+\frac {\left (-25 x +100\right ) {\mathrm e}^{-\frac {4 x^{2}}{25}}}{225}\) | \(28\) |
default | \(2 x -\frac {8}{9 x}-x^{3}+\frac {4 \,{\mathrm e}^{-\frac {4 x^{2}}{25}}}{9}-\frac {x \,{\mathrm e}^{-\frac {4 x^{2}}{25}}}{9}\) | \(34\) |
norman | \(\frac {\left (\frac {4 x}{9}-\frac {x^{2}}{9}+2 x^{2} {\mathrm e}^{\frac {4 x^{2}}{25}}-{\mathrm e}^{\frac {4 x^{2}}{25}} x^{4}-\frac {8 \,{\mathrm e}^{\frac {4 x^{2}}{25}}}{9}\right ) {\mathrm e}^{-\frac {4 x^{2}}{25}}}{x}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 31, normalized size = 0.94 \begin {gather*} -x^{3} - \frac {1}{9} \, x e^{\left (-\frac {4}{25} \, x^{2}\right )} + 2 \, x - \frac {8}{9 \, x} + \frac {4}{9} \, e^{\left (-\frac {4}{25} \, x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 32, normalized size = 0.97 \begin {gather*} \frac {4\,{\mathrm {e}}^{-\frac {4\,x^2}{25}}}{9}-x\,\left (\frac {{\mathrm {e}}^{-\frac {4\,x^2}{25}}}{9}-2\right )-\frac {8}{9\,x}-x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 24, normalized size = 0.73 \begin {gather*} - x^{3} + 2 x + \frac {\left (4 - x\right ) e^{- \frac {4 x^{2}}{25}}}{9} - \frac {8}{9 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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