Optimal. Leaf size=23 \[ 13-2 x+\frac {1}{\log ^2\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \]
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Rubi [F] time = 15.04, 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^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 \left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right )+2 x \log (x) \left (4 x+\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )\right )}{x \left (2-x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx\\ &=\int \left (-2-\frac {4 \left (2-x^2+2 x^2 \log (x)+2 \log \left (-2+x^2\right )-x^2 \log \left (-2+x^2\right )\right )}{x \left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx\\ &=-2 x-4 \int \frac {2-x^2+2 x^2 \log (x)+2 \log \left (-2+x^2\right )-x^2 \log \left (-2+x^2\right )}{x \left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx\\ &=-2 x-4 \int \left (\frac {-2+x^2-2 x^2 \log (x)-2 \log \left (-2+x^2\right )+x^2 \log \left (-2+x^2\right )}{2 x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {2 x-x^3+2 x^3 \log (x)+2 x \log \left (-2+x^2\right )-x^3 \log \left (-2+x^2\right )}{2 \left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx\\ &=-2 x-2 \int \frac {-2+x^2-2 x^2 \log (x)-2 \log \left (-2+x^2\right )+x^2 \log \left (-2+x^2\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-2 \int \frac {2 x-x^3+2 x^3 \log (x)+2 x \log \left (-2+x^2\right )-x^3 \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx\\ &=-2 x-2 \int \left (\frac {2 x^3}{\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {2 x}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}-\frac {x^3}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {2 x \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}-\frac {x^3 \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx-2 \int \frac {-2 x^2 \log (x)+\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx\\ &=-2 x-2 \int \left (-\frac {2 x}{\left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}-\frac {2}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {x}{\log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}-\frac {2 \log \left (-2+x^2\right )}{x \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}+\frac {x \log \left (-2+x^2\right )}{\log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )}\right ) \, dx+2 \int \frac {x^3}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx+2 \int \frac {x^3 \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-4 \int \frac {x^3}{\left (-2+x^2\right ) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-4 \int \frac {x}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx-4 \int \frac {x \log \left (-2+x^2\right )}{\left (-2+x^2\right ) \log (x) \left (1+\log \left (-2+x^2\right )\right ) \log ^3\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.15, size = 22, normalized size = 0.96 \begin {gather*} -2 x+\frac {1}{\log ^2\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 59, normalized size = 2.57 \begin {gather*} -\frac {2 \, x \log \left (\frac {\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1}{\log \relax (x)^{2}}\right )^{2} - 1}{\log \left (\frac {\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1}{\log \relax (x)^{2}}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 69.36, size = 393, normalized size = 17.09 \begin {gather*} -2 \, x + \frac {x^{2} \log \left (x^{2} - 2\right ) - 2 \, x^{2} \log \relax (x) + x^{2} - 2 \, \log \left (x^{2} - 2\right ) - 2}{x^{2} \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} - 2 \, x^{2} \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \relax (x)^{2}\right ) + 4 \, x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \relax (x) \log \left (\log \relax (x)^{2}\right ) + x^{2} \log \left (x^{2} - 2\right ) \log \left (\log \relax (x)^{2}\right )^{2} - 2 \, x^{2} \log \relax (x) \log \left (\log \relax (x)^{2}\right )^{2} - 2 \, x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} \log \relax (x) + x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} - 2 \, x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \relax (x)^{2}\right ) + x^{2} \log \left (\log \relax (x)^{2}\right )^{2} - 2 \, \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} + 4 \, \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \relax (x)^{2}\right ) - 2 \, \log \left (x^{2} - 2\right ) \log \left (\log \relax (x)^{2}\right )^{2} - 2 \, \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} + 4 \, \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \relax (x)^{2}\right ) - 2 \, \log \left (\log \relax (x)^{2}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.88, size = 290, normalized size = 12.61
method | result | size |
risch | \(-2 x -\frac {4}{\left (\pi \mathrm {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )\right )^{2} \mathrm {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )\right ) \mathrm {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (x^{2}-2\right )+1\right )^{2}}{\ln \relax (x )^{2}}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \left (x^{2}-2\right )+1\right )^{2}}{\ln \relax (x )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right )+\pi \mathrm {csgn}\left (\frac {i \left (\ln \left (x^{2}-2\right )+1\right )^{2}}{\ln \relax (x )^{2}}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i \left (\ln \left (x^{2}-2\right )+1\right )^{2}}{\ln \relax (x )^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )^{2}}\right )-\pi \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \ln \relax (x )^{2}\right )^{3}-4 i \ln \left (\ln \relax (x )\right )+4 i \ln \left (\ln \left (x^{2}-2\right )+1\right )\right )^{2}}\) | \(290\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 74, normalized size = 3.22 \begin {gather*} -\frac {8 \, x \log \left (\log \left (x^{2} - 2\right ) + 1\right )^{2} - 16 \, x \log \left (\log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \relax (x)\right ) + 8 \, x \log \left (\log \relax (x)\right )^{2} - 1}{4 \, {\left (\log \left (\log \left (x^{2} - 2\right ) + 1\right )^{2} - 2 \, \log \left (\log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \relax (x)\right ) + \log \left (\log \relax (x)\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.53, size = 30, normalized size = 1.30 \begin {gather*} \frac {1}{{\ln \left (\frac {{\ln \left (x^2-2\right )}^2+2\,\ln \left (x^2-2\right )+1}{{\ln \relax (x)}^2}\right )}^2}-2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.07, size = 31, normalized size = 1.35 \begin {gather*} - 2 x + \frac {1}{\log {\left (\frac {\log {\left (x^{2} - 2 \right )}^{2} + 2 \log {\left (x^{2} - 2 \right )} + 1}{\log {\relax (x )}^{2}} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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