Optimal. Leaf size=31 \[ -x+9 x^2+\frac {3 e^{3-x}}{x (4 x-2 \log (x))} \]
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Rubi [A] time = 1.60, antiderivative size = 47, normalized size of antiderivative = 1.52, number of steps used = 5, number of rules used = 4, integrand size = 111, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6741, 12, 6742, 2288} \begin {gather*} \frac {3 e^{3-x} \left (2 x^2-x \log (x)\right )}{2 x^2 (2 x-\log (x))^2}+\frac {1}{36} (1-18 x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2288
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (e^3 \left (3-12 x-6 x^2\right )+e^x \left (-8 x^4+144 x^5\right )+\left (e^3 (3+3 x)+e^x \left (8 x^3-144 x^4\right )\right ) \log (x)+e^x \left (-2 x^2+36 x^3\right ) \log ^2(x)\right )}{2 x^2 (2 x-\log (x))^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{-x} \left (e^3 \left (3-12 x-6 x^2\right )+e^x \left (-8 x^4+144 x^5\right )+\left (e^3 (3+3 x)+e^x \left (8 x^3-144 x^4\right )\right ) \log (x)+e^x \left (-2 x^2+36 x^3\right ) \log ^2(x)\right )}{x^2 (2 x-\log (x))^2} \, dx\\ &=\frac {1}{2} \int \left (2 (-1+18 x)-\frac {3 e^{3-x} \left (-1+4 x+2 x^2-\log (x)-x \log (x)\right )}{x^2 (2 x-\log (x))^2}\right ) \, dx\\ &=\frac {1}{36} (1-18 x)^2-\frac {3}{2} \int \frac {e^{3-x} \left (-1+4 x+2 x^2-\log (x)-x \log (x)\right )}{x^2 (2 x-\log (x))^2} \, dx\\ &=\frac {1}{36} (1-18 x)^2+\frac {3 e^{3-x} \left (2 x^2-x \log (x)\right )}{2 x^2 (2 x-\log (x))^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 33, normalized size = 1.06 \begin {gather*} \frac {1}{2} \left (-2 x+18 x^2-\frac {3 e^{3-x}}{x (-2 x+\log (x))}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 56, normalized size = 1.81 \begin {gather*} -\frac {2 \, {\left (9 \, x^{3} - x^{2}\right )} e^{x} \log \relax (x) - 4 \, {\left (9 \, x^{4} - x^{3}\right )} e^{x} - 3 \, e^{3}}{2 \, {\left (2 \, x^{2} e^{x} - x e^{x} \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 48, normalized size = 1.55 \begin {gather*} \frac {36 \, x^{4} - 18 \, x^{3} \log \relax (x) - 4 \, x^{3} + 2 \, x^{2} \log \relax (x) + 3 \, e^{\left (-x + 3\right )}}{2 \, {\left (2 \, x^{2} - x \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 31, normalized size = 1.00
method | result | size |
risch | \(9 x^{2}-x +\frac {3 \,{\mathrm e}^{3-x}}{2 \left (2 x -\ln \relax (x )\right ) x}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 49, normalized size = 1.58 \begin {gather*} \frac {36 \, x^{4} - 4 \, x^{3} - 2 \, {\left (9 \, x^{3} - x^{2}\right )} \log \relax (x) + 3 \, e^{\left (-x + 3\right )}}{2 \, {\left (2 \, x^{2} - x \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.49, size = 105, normalized size = 3.39 \begin {gather*} \frac {\frac {3\,{\mathrm {e}}^{-x}\,\left (2\,{\mathrm {e}}^3\,x^2+4\,{\mathrm {e}}^3\,x-{\mathrm {e}}^3\right )}{2\,x\,\left (2\,x-1\right )}-\frac {3\,{\mathrm {e}}^{-x}\,\ln \relax (x)\,\left ({\mathrm {e}}^3+x\,{\mathrm {e}}^3\right )}{2\,x\,\left (2\,x-1\right )}}{2\,x-\ln \relax (x)}-x+9\,x^2+\frac {{\mathrm {e}}^{-x}\,\left (\frac {3\,{\mathrm {e}}^3}{4}+\frac {3\,x\,{\mathrm {e}}^3}{4}\right )}{\frac {x}{2}-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 26, normalized size = 0.84 \begin {gather*} 9 x^{2} - x + \frac {3 e^{3} e^{- x}}{4 x^{2} - 2 x \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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