Optimal. Leaf size=29 \[ (-4+2 x)^2 \left (-e^{x^2}+x \left (4-\frac {1}{9} (3+x)^2\right )\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.97, number of steps used = 10, number of rules used = 5, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2226, 2204, 2209, 2212} \begin {gather*} -\frac {4 x^5}{9}-\frac {8 x^4}{9}+\frac {188 x^3}{9}-4 e^{x^2} x^2-\frac {176 x^2}{3}+16 e^{x^2} x-16 e^{x^2}+48 x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2204
Rule 2209
Rule 2212
Rule 2226
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \left (432-1056 x+564 x^2-32 x^3-20 x^4+e^{x^2} \left (144-360 x+288 x^2-72 x^3\right )\right ) \, dx\\ &=48 x-\frac {176 x^2}{3}+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}+\frac {1}{9} \int e^{x^2} \left (144-360 x+288 x^2-72 x^3\right ) \, dx\\ &=48 x-\frac {176 x^2}{3}+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}+\frac {1}{9} \int \left (144 e^{x^2}-360 e^{x^2} x+288 e^{x^2} x^2-72 e^{x^2} x^3\right ) \, dx\\ &=48 x-\frac {176 x^2}{3}+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}-8 \int e^{x^2} x^3 \, dx+16 \int e^{x^2} \, dx+32 \int e^{x^2} x^2 \, dx-40 \int e^{x^2} x \, dx\\ &=-20 e^{x^2}+48 x+16 e^{x^2} x-\frac {176 x^2}{3}-4 e^{x^2} x^2+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}+8 \sqrt {\pi } \text {erfi}(x)+8 \int e^{x^2} x \, dx-16 \int e^{x^2} \, dx\\ &=-16 e^{x^2}+48 x+16 e^{x^2} x-\frac {176 x^2}{3}-4 e^{x^2} x^2+\frac {188 x^3}{9}-\frac {8 x^4}{9}-\frac {4 x^5}{9}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.33, size = 27, normalized size = 0.93 \begin {gather*} -\frac {4}{9} (-2+x)^2 \left (9 e^{x^2}+x \left (-27+6 x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 38, normalized size = 1.31 \begin {gather*} -\frac {4}{9} \, x^{5} - \frac {8}{9} \, x^{4} + \frac {188}{9} \, x^{3} - \frac {176}{3} \, x^{2} - 4 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 48 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 38, normalized size = 1.31 \begin {gather*} -\frac {4}{9} \, x^{5} - \frac {8}{9} \, x^{4} + \frac {188}{9} \, x^{3} - \frac {176}{3} \, x^{2} - 4 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 48 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 41, normalized size = 1.41
method | result | size |
risch | \(\frac {\left (-36 x^{2}+144 x -144\right ) {\mathrm e}^{x^{2}}}{9}-\frac {4 x^{5}}{9}-\frac {8 x^{4}}{9}+\frac {188 x^{3}}{9}-\frac {176 x^{2}}{3}+48 x\) | \(41\) |
default | \(48 x -\frac {176 x^{2}}{3}+\frac {188 x^{3}}{9}-\frac {8 x^{4}}{9}-\frac {4 x^{5}}{9}-16 \,{\mathrm e}^{x^{2}}+16 \,{\mathrm e}^{x^{2}} x -4 x^{2} {\mathrm e}^{x^{2}}\) | \(47\) |
norman | \(48 x -\frac {176 x^{2}}{3}+\frac {188 x^{3}}{9}-\frac {8 x^{4}}{9}-\frac {4 x^{5}}{9}-16 \,{\mathrm e}^{x^{2}}+16 \,{\mathrm e}^{x^{2}} x -4 x^{2} {\mathrm e}^{x^{2}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 38, normalized size = 1.31 \begin {gather*} -\frac {4}{9} \, x^{5} - \frac {8}{9} \, x^{4} + \frac {188}{9} \, x^{3} - \frac {176}{3} \, x^{2} - 4 \, {\left (x^{2} - 4 \, x + 4\right )} e^{\left (x^{2}\right )} + 48 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 25, normalized size = 0.86 \begin {gather*} -\frac {4\,{\left (x-2\right )}^2\,\left (9\,{\mathrm {e}}^{x^2}-27\,x+6\,x^2+x^3\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 44, normalized size = 1.52 \begin {gather*} - \frac {4 x^{5}}{9} - \frac {8 x^{4}}{9} + \frac {188 x^{3}}{9} - \frac {176 x^{2}}{3} + 48 x + \left (- 4 x^{2} + 16 x - 16\right ) e^{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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