Optimal. Leaf size=21 \[ -3+\frac {3 e}{x}-\frac {x^2}{\frac {1}{2}+\log (x)} \]
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Rubi [A] time = 0.35, antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 6, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6688, 2561, 6742, 2306, 2309, 2178} \begin {gather*} \frac {3 e}{x}-\frac {2 x^2}{2 \log (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2309
Rule 2561
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 e-4 \left (3 e+2 x^3\right ) \log (x)-12 e \log ^2(x)}{(x+2 x \log (x))^2} \, dx\\ &=\int \frac {-3 e-4 \left (3 e+2 x^3\right ) \log (x)-12 e \log ^2(x)}{x^2 (1+2 \log (x))^2} \, dx\\ &=\int \left (-\frac {3 e}{x^2}+\frac {4 x}{(1+2 \log (x))^2}-\frac {4 x}{1+2 \log (x)}\right ) \, dx\\ &=\frac {3 e}{x}+4 \int \frac {x}{(1+2 \log (x))^2} \, dx-4 \int \frac {x}{1+2 \log (x)} \, dx\\ &=\frac {3 e}{x}-\frac {2 x^2}{1+2 \log (x)}+4 \int \frac {x}{1+2 \log (x)} \, dx-4 \operatorname {Subst}\left (\int \frac {e^{2 x}}{1+2 x} \, dx,x,\log (x)\right )\\ &=\frac {3 e}{x}-\frac {2 \text {Ei}(1+2 \log (x))}{e}-\frac {2 x^2}{1+2 \log (x)}+4 \operatorname {Subst}\left (\int \frac {e^{2 x}}{1+2 x} \, dx,x,\log (x)\right )\\ &=\frac {3 e}{x}-\frac {2 x^2}{1+2 \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 20, normalized size = 0.95 \begin {gather*} \frac {3 e}{x}-\frac {2 x^2}{1+2 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 27, normalized size = 1.29 \begin {gather*} -\frac {2 \, x^{3} - 6 \, e \log \relax (x) - 3 \, e}{2 \, x \log \relax (x) + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 27, normalized size = 1.29 \begin {gather*} -\frac {2 \, x^{3} - 6 \, e \log \relax (x) - 3 \, e}{2 \, x \log \relax (x) + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 22, normalized size = 1.05
method | result | size |
risch | \(\frac {3 \,{\mathrm e}}{x}-\frac {2 x^{2}}{1+2 \ln \relax (x )}\) | \(22\) |
norman | \(\frac {-2 x^{3}+6 \,{\mathrm e} \ln \relax (x )+3 \,{\mathrm e}}{x \left (1+2 \ln \relax (x )\right )}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 27, normalized size = 1.29 \begin {gather*} -\frac {2 \, x^{3} - 6 \, e \log \relax (x) - 3 \, e}{2 \, x \log \relax (x) + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 21, normalized size = 1.00 \begin {gather*} \frac {3\,\mathrm {e}}{x}-\frac {2\,x^2}{2\,\ln \relax (x)+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 17, normalized size = 0.81 \begin {gather*} - \frac {2 x^{2}}{2 \log {\relax (x )} + 1} + \frac {3 e}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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