Optimal. Leaf size=33 \[ \frac {4 e^x \left (2-e^{-x+\frac {5}{3 x^2 \log (x)}}-\log (x)\right )}{x^2} \]
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Rubi [A] time = 0.92, antiderivative size = 66, normalized size of antiderivative = 2.00, number of steps used = 5, number of rules used = 3, integrand size = 98, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {12, 6742, 2288} \begin {gather*} \frac {4 e^x (2 x-x \log (x))}{x^3}-\frac {4 e^{\frac {5}{3 x^2 \log (x)}} (2 \log (x)+1)}{x^5 \left (\frac {1}{x^3 \log ^2(x)}+\frac {2}{x^3 \log (x)}\right ) \log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2288
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^x \left (-60 x^2+24 x^3\right ) \log ^2(x)+e^x \left (24 x^2-12 x^3\right ) \log ^3(x)+e^{\frac {5-3 x^3 \log (x)}{3 x^2 \log (x)}} \left (20 e^x+40 e^x \log (x)+24 e^x x^2 \log ^2(x)\right )}{x^5 \log ^2(x)} \, dx\\ &=\frac {1}{3} \int \left (-\frac {12 e^x (5-2 x-2 \log (x)+x \log (x))}{x^3}+\frac {4 e^{\frac {5}{3 x^2 \log (x)}} \left (5+10 \log (x)+6 x^2 \log ^2(x)\right )}{x^5 \log ^2(x)}\right ) \, dx\\ &=\frac {4}{3} \int \frac {e^{\frac {5}{3 x^2 \log (x)}} \left (5+10 \log (x)+6 x^2 \log ^2(x)\right )}{x^5 \log ^2(x)} \, dx-4 \int \frac {e^x (5-2 x-2 \log (x)+x \log (x))}{x^3} \, dx\\ &=-\frac {4 e^{\frac {5}{3 x^2 \log (x)}} (1+2 \log (x))}{x^5 \left (\frac {1}{x^3 \log ^2(x)}+\frac {2}{x^3 \log (x)}\right ) \log ^2(x)}+\frac {4 e^x (2 x-x \log (x))}{x^3}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 30, normalized size = 0.91 \begin {gather*} -\frac {4 \left (-2 e^x+e^{\frac {5}{3 x^2 \log (x)}}+e^x \log (x)\right )}{x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 36, normalized size = 1.09 \begin {gather*} -\frac {4 \, {\left (e^{x} \log \relax (x) + e^{\left (x - \frac {3 \, x^{3} \log \relax (x) - 5}{3 \, x^{2} \log \relax (x)}\right )} - 2 \, e^{x}\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.93, size = 25, normalized size = 0.76 \begin {gather*} -\frac {4 \, {\left (e^{x} \log \relax (x) - 2 \, e^{x} + e^{\left (\frac {5}{3 \, x^{2} \log \relax (x)}\right )}\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 33, normalized size = 1.00
method | result | size |
risch | \(-\frac {4 \,{\mathrm e}^{x} \ln \relax (x )}{x^{2}}+\frac {8 \,{\mathrm e}^{x}}{x^{2}}-\frac {4 \,{\mathrm e}^{\frac {5}{3 x^{2} \ln \relax (x )}}}{x^{2}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 32, normalized size = 0.97 \begin {gather*} \frac {8\,{\mathrm {e}}^x}{x^2}-\frac {4\,{\mathrm {e}}^{\frac {5}{3\,x^2\,\ln \relax (x)}}}{x^2}-\frac {4\,{\mathrm {e}}^x\,\ln \relax (x)}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 39, normalized size = 1.18 \begin {gather*} \frac {\left (8 - 4 \log {\relax (x )}\right ) e^{x}}{x^{2}} - \frac {4 e^{x} e^{\frac {- x^{3} \log {\relax (x )} + \frac {5}{3}}{x^{2} \log {\relax (x )}}}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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