Optimal. Leaf size=24 \[ \frac {1}{-e^x+\frac {\log \left (x^2\right )}{2}+\log \left (-e^x+x\right )} \]
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Rubi [A] time = 0.57, antiderivative size = 26, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 2, integrand size = 154, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6688, 6686} \begin {gather*} -\frac {2}{-\log \left (x^2\right )+2 e^x-2 \log \left (x-e^x\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6686
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 x+4 e^{2 x} x-4 e^x \left (1+x+x^2\right )}{\left (e^x-x\right ) x \left (2 e^x-\log \left (x^2\right )-2 \log \left (-e^x+x\right )\right )^2} \, dx\\ &=-\frac {2}{2 e^x-\log \left (x^2\right )-2 \log \left (-e^x+x\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 24, normalized size = 1.00 \begin {gather*} \frac {2}{-2 e^x+\log \left (x^2\right )+2 \log \left (-e^x+x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2}{2 \, e^{x} - \log \left (x^{2}\right ) - 2 \, \log \left (x - e^{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {4 \, {\left (x e^{\left (2 \, x\right )} - {\left (x^{2} + x + 1\right )} e^{x} + 2 \, x\right )}}{4 \, x^{2} e^{\left (2 \, x\right )} + {\left (x^{2} - x e^{x}\right )} \log \left (x^{2}\right )^{2} + 4 \, {\left (x^{2} - x e^{x}\right )} \log \left (x - e^{x}\right )^{2} - 4 \, x e^{\left (3 \, x\right )} - 4 \, {\left (x^{2} e^{x} - x e^{\left (2 \, x\right )}\right )} \log \left (x^{2}\right ) - 4 \, {\left (2 \, x^{2} e^{x} - 2 \, x e^{\left (2 \, x\right )} - {\left (x^{2} - x e^{x}\right )} \log \left (x^{2}\right )\right )} \log \left (x - e^{x}\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 71, normalized size = 2.96
method | result | size |
risch | \(\frac {4 i}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-4 i {\mathrm e}^{x}+4 i \ln \relax (x )+4 i \ln \left (x -{\mathrm e}^{x}\right )}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 20, normalized size = 0.83 \begin {gather*} -\frac {1}{e^{x} - \log \left (x - e^{x}\right ) - \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.25, size = 22, normalized size = 0.92 \begin {gather*} \frac {2}{\ln \left (x^2\right )+2\,\ln \left (x-{\mathrm {e}}^x\right )-2\,{\mathrm {e}}^x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 19, normalized size = 0.79 \begin {gather*} \frac {2}{- 2 e^{x} + \log {\left (x^{2} \right )} + 2 \log {\left (x - e^{x} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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