Optimal. Leaf size=24 \[ \frac {4 \left (-2+2 e^2\right )}{x^2 (e+x)}+\frac {3}{\log (x)} \]
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Rubi [A] time = 0.37, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {1594, 27, 6688, 74, 2302, 30} \begin {gather*} \frac {3}{\log (x)}-\frac {8 \left (1-e^2\right )}{x^2 (x+e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 30
Rule 74
Rule 1594
Rule 2302
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 e^2 x^2-6 e x^3-3 x^4+\left (16 e+2 e^2 (-8 e-12 x)+24 x\right ) \log ^2(x)}{x^3 \left (e^2+2 e x+x^2\right ) \log ^2(x)} \, dx\\ &=\int \frac {-3 e^2 x^2-6 e x^3-3 x^4+\left (16 e+2 e^2 (-8 e-12 x)+24 x\right ) \log ^2(x)}{x^3 (e+x)^2 \log ^2(x)} \, dx\\ &=\int \left (-\frac {8 \left (-1+e^2\right ) (2 e+3 x)}{x^3 (e+x)^2}-\frac {3}{x \log ^2(x)}\right ) \, dx\\ &=-\left (3 \int \frac {1}{x \log ^2(x)} \, dx\right )+\left (8 \left (1-e^2\right )\right ) \int \frac {2 e+3 x}{x^3 (e+x)^2} \, dx\\ &=-\frac {8 \left (1-e^2\right )}{x^2 (e+x)}-3 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=-\frac {8 \left (1-e^2\right )}{x^2 (e+x)}+\frac {3}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 0.92 \begin {gather*} \frac {8 \left (-1+e^2\right )}{x^2 (e+x)}+\frac {3}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 38, normalized size = 1.58 \begin {gather*} \frac {3 \, x^{3} + 3 \, x^{2} e + 8 \, {\left (e^{2} - 1\right )} \log \relax (x)}{{\left (x^{3} + x^{2} e\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 41, normalized size = 1.71 \begin {gather*} \frac {3 \, x^{3} + 3 \, x^{2} e + 8 \, e^{2} \log \relax (x) - 8 \, \log \relax (x)}{x^{3} \log \relax (x) + x^{2} e \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 23, normalized size = 0.96
method | result | size |
risch | \(\frac {3}{\ln \relax (x )}+\frac {8 \,{\mathrm e}^{2}-8}{x^{2} \left (x +{\mathrm e}\right )}\) | \(23\) |
default | \(\frac {3}{\ln \relax (x )}+\frac {8 \,{\mathrm e}^{2}-8}{x^{2} \left (x +{\mathrm e}\right )}\) | \(24\) |
norman | \(\frac {\left (8 \,{\mathrm e}^{2}-8\right ) \ln \relax (x )+3 x^{3}+3 x^{2} {\mathrm e}}{x^{2} \left (x +{\mathrm e}\right ) \ln \relax (x )}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 38, normalized size = 1.58 \begin {gather*} \frac {3 \, x^{3} + 3 \, x^{2} e + 8 \, {\left (e^{2} - 1\right )} \log \relax (x)}{{\left (x^{3} + x^{2} e\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.44, size = 44, normalized size = 1.83 \begin {gather*} \frac {8\,{\mathrm {e}}^2-8}{x^2\,\left (x+\mathrm {e}\right )}+\frac {3\,x^3+3\,\mathrm {e}\,x^2}{x^2\,\ln \relax (x)\,\left (x+\mathrm {e}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 20, normalized size = 0.83 \begin {gather*} \frac {3}{\log {\relax (x )}} + \frac {-8 + 8 e^{2}}{x^{3} + e x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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