Optimal. Leaf size=27 \[ \frac {1}{x \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \]
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Rubi [F] time = 2.55, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-8+4 x-26 x^2+28 x^3-6 x^4+\left (4-4 x-3 x^2+4 x^3-x^4\right ) \log \left (\frac {e^{\frac {x}{-2+x}} \left (1-2 x^2+x^4\right )}{x}\right )}{\left (-4 x^2+4 x^3+3 x^4-4 x^5+x^6\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (1-2 x^2+x^4\right )}{x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8-4 x+26 x^2-28 x^3+6 x^4-\left (4-4 x-3 x^2+4 x^3-x^4\right ) \log \left (\frac {e^{\frac {x}{-2+x}} \left (1-2 x^2+x^4\right )}{x}\right )}{(2-x)^2 x^2 \left (1-x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (1-2 x^2+x^4\right )}{x}\right )} \, dx\\ &=\int \left (-\frac {2 \left (4-2 x+13 x^2-14 x^3+3 x^4\right )}{(-2+x)^2 x^2 \left (-1+x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}-\frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {4-2 x+13 x^2-14 x^3+3 x^4}{(-2+x)^2 x^2 \left (-1+x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\right )-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ &=-\left (2 \int \left (-\frac {1}{(-2+x)^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}+\frac {1}{2 (-2+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}-\frac {1}{x^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}-\frac {1}{2 x \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}+\frac {4}{\left (-1+x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}\right ) \, dx\right )-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ &=2 \int \frac {1}{(-2+x)^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+2 \int \frac {1}{x^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-8 \int \frac {1}{\left (-1+x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{(-2+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+\int \frac {1}{x \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ &=2 \int \frac {1}{(-2+x)^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+2 \int \frac {1}{x^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-8 \int \left (\frac {1}{2 (-1+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}-\frac {1}{2 (1+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}\right ) \, dx-\int \frac {1}{(-2+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+\int \frac {1}{x \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ &=2 \int \frac {1}{(-2+x)^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+2 \int \frac {1}{x^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-4 \int \frac {1}{(-1+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+4 \int \frac {1}{(1+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{(-2+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+\int \frac {1}{x \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 27, normalized size = 1.00 \begin {gather*} \frac {1}{x \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 29, normalized size = 1.07 \begin {gather*} \frac {1}{x \log \left (\frac {{\left (x^{4} - 2 \, x^{2} + 1\right )} e^{\left (\frac {x}{x - 2}\right )}}{x}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.31, size = 331, normalized size = 12.26 \begin {gather*} \frac {3 \, x^{6} - 26 \, x^{5} + 81 \, x^{4} - 110 \, x^{3} + 64 \, x^{2} - 24 \, x + 16}{3 \, x^{7} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} + 6 \, x^{7} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) - 26 \, x^{6} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} + 3 \, x^{7} - 40 \, x^{6} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) + 81 \, x^{5} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} - 14 \, x^{6} + 82 \, x^{5} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) - 110 \, x^{4} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} + 13 \, x^{5} - 56 \, x^{4} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) + 64 \, x^{3} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} - 2 \, x^{4} + 16 \, x^{3} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) - 24 \, x^{2} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} + 4 \, x^{3} - 16 \, x^{2} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) + 16 \, x \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.38, size = 393, normalized size = 14.56
method | result | size |
risch | \(-\frac {4}{x \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-1\right )^{2} {\mathrm e}^{\frac {x}{x -2}}}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-1\right )^{2} {\mathrm e}^{\frac {x}{x -2}}}{x}\right )^{2}+\pi \mathrm {csgn}\left (i \left (x^{2}-1\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (x^{2}-1\right )\right ) \mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right )-\pi \,\mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-1\right )^{2} {\mathrm e}^{\frac {x}{x -2}}}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x^{2}-1\right )^{2} {\mathrm e}^{\frac {x}{x -2}}}{x}\right )^{3}-2 i \ln \relax (x )+4 i \ln \left (x^{2}-1\right )+2 i \ln \left ({\mathrm e}^{\frac {x}{x -2}}\right )\right )^{2}}\) | \(393\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.94, size = 165, normalized size = 6.11 \begin {gather*} \frac {x^{2} - 4 \, x + 4}{x^{3} + 4 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (x + 1\right )^{2} + 4 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (x - 1\right )^{2} + {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \relax (x)^{2} + 4 \, {\left (x^{3} - 2 \, x^{2} + 2 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (x - 1\right ) - {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \relax (x)\right )} \log \left (x + 1\right ) + 4 \, {\left (x^{3} - 2 \, x^{2} - {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \relax (x)\right )} \log \left (x - 1\right ) - 2 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.33, size = 29, normalized size = 1.07 \begin {gather*} \frac {1}{x\,{\ln \left (\frac {{\mathrm {e}}^{\frac {x}{x-2}}\,\left (x^4-2\,x^2+1\right )}{x}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{x \log {\left (\frac {\left (x^{4} - 2 x^{2} + 1\right ) e^{\frac {x}{x - 2}}}{x} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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