Optimal. Leaf size=18 \[ \frac {1}{-2-\frac {4 x}{2-e^2}+\log (x)} \]
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Rubi [A] time = 0.21, antiderivative size = 35, normalized size of antiderivative = 1.94, number of steps used = 4, number of rules used = 4, integrand size = 106, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6, 6688, 12, 6686} \begin {gather*} -\frac {2-e^2}{4 x-\left (2-e^2\right ) \log (x)+2 \left (2-e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 6686
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-e^4+e^2 (4-4 x)+8 x}{\left (16+4 e^4\right ) x+32 x^2+16 x^3+e^2 \left (-16 x-16 x^2\right )+\left (-16 x-4 e^4 x-16 x^2+e^2 \left (16 x+8 x^2\right )\right ) \log (x)+\left (4 x-4 e^2 x+e^4 x\right ) \log ^2(x)} \, dx\\ &=\int \frac {\left (2-e^2\right ) \left (-2+e^2+4 x\right )}{x \left (4 \left (1-\frac {e^2}{2}\right )+4 x+\left (-2+e^2\right ) \log (x)\right )^2} \, dx\\ &=\left (2-e^2\right ) \int \frac {-2+e^2+4 x}{x \left (4 \left (1-\frac {e^2}{2}\right )+4 x+\left (-2+e^2\right ) \log (x)\right )^2} \, dx\\ &=-\frac {2-e^2}{2 \left (2-e^2\right )+4 x-\left (2-e^2\right ) \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 1.44 \begin {gather*} \frac {-2+e^2}{4-2 e^2+4 x+\left (-2+e^2\right ) \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 23, normalized size = 1.28 \begin {gather*} \frac {e^{2} - 2}{{\left (e^{2} - 2\right )} \log \relax (x) + 4 \, x - 2 \, e^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 25, normalized size = 1.39 \begin {gather*} \frac {e^{2} - 2}{e^{2} \log \relax (x) + 4 \, x - 2 \, e^{2} - 2 \, \log \relax (x) + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 26, normalized size = 1.44
method | result | size |
norman | \(\frac {{\mathrm e}^{2}-2}{{\mathrm e}^{2} \ln \relax (x )-2 \,{\mathrm e}^{2}-2 \ln \relax (x )+4 x +4}\) | \(26\) |
risch | \(\frac {{\mathrm e}^{2}}{{\mathrm e}^{2} \ln \relax (x )-2 \,{\mathrm e}^{2}-2 \ln \relax (x )+4 x +4}-\frac {2}{{\mathrm e}^{2} \ln \relax (x )-2 \,{\mathrm e}^{2}-2 \ln \relax (x )+4 x +4}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 23, normalized size = 1.28 \begin {gather*} \frac {e^{2} - 2}{{\left (e^{2} - 2\right )} \log \relax (x) + 4 \, x - 2 \, e^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 25, normalized size = 1.39 \begin {gather*} \frac {{\mathrm {e}}^2-2}{4\,x-2\,{\mathrm {e}}^2-2\,\ln \relax (x)+{\mathrm {e}}^2\,\ln \relax (x)+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 22, normalized size = 1.22 \begin {gather*} \frac {-2 + e^{2}}{4 x + \left (-2 + e^{2}\right ) \log {\relax (x )} - 2 e^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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