Optimal. Leaf size=239 \[ -\frac {\text {Li}_2\left (1-\frac {2 \sqrt {2}}{x+\sqrt {2}}\right )}{\sqrt {2}}+\frac {\text {Li}_2\left (\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}+1\right )}{2 \sqrt {2}}+\frac {\text {Li}_2\left (1-\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )}{2 \sqrt {2}}+\frac {\log \left (1-x^2\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}+\sqrt {2} \log \left (\frac {2 \sqrt {2}}{x+\sqrt {2}}\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {\log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\log \left (\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.29, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {206, 2470, 12, 5992, 5920, 2402, 2315, 2447} \[ -\frac {\text {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{x+\sqrt {2}}\right )}{\sqrt {2}}+\frac {\text {PolyLog}\left (2,\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}+1\right )}{2 \sqrt {2}}+\frac {\text {PolyLog}\left (2,1-\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )}{2 \sqrt {2}}+\frac {\log \left (1-x^2\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}+\sqrt {2} \log \left (\frac {2 \sqrt {2}}{x+\sqrt {2}}\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {\log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\log \left (\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2315
Rule 2402
Rule 2447
Rule 2470
Rule 5920
Rule 5992
Rubi steps
\begin {align*} \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx &=\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}+2 \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2} \left (1-x^2\right )} \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}+\sqrt {2} \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{1-x^2} \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}+\sqrt {2} \int \left (-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 (-1+x)}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{2 (1+x)}\right ) \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}-\frac {\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{-1+x} \, dx}{\sqrt {2}}-\frac {\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{1+x} \, dx}{\sqrt {2}}\\ &=\sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}-2 \left (\frac {1}{2} \int \frac {\log \left (\frac {2}{1+\frac {x}{\sqrt {2}}}\right )}{1-\frac {x^2}{2}} \, dx\right )+\frac {1}{2} \int \frac {\log \left (\frac {\sqrt {2} (-1+x)}{\left (1-\frac {1}{\sqrt {2}}\right ) \left (1+\frac {x}{\sqrt {2}}\right )}\right )}{1-\frac {x^2}{2}} \, dx+\frac {1}{2} \int \frac {\log \left (\frac {\sqrt {2} (1+x)}{\left (1+\frac {1}{\sqrt {2}}\right ) \left (1+\frac {x}{\sqrt {2}}\right )}\right )}{1-\frac {x^2}{2}} \, dx\\ &=\sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}+\frac {\text {Li}_2\left (1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}}+\frac {\text {Li}_2\left (1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}}-2 \frac {\operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {x}{\sqrt {2}}}\right )}{\sqrt {2}}\\ &=\sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}-\frac {\text {Li}_2\left (1-\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )}{\sqrt {2}}+\frac {\text {Li}_2\left (1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}}+\frac {\text {Li}_2\left (1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 248, normalized size = 1.04 \[ \frac {\text {Li}_2\left (\frac {x-1}{-1-\sqrt {2}}\right )+\log \left (1-\frac {x-1}{-1-\sqrt {2}}\right ) \log (x-1)}{2 \sqrt {2}}-\frac {\text {Li}_2\left (\frac {x-1}{-1+\sqrt {2}}\right )+\log \left (1-\frac {x-1}{\sqrt {2}-1}\right ) \log (x-1)}{2 \sqrt {2}}+\frac {\text {Li}_2\left (\frac {x+1}{1-\sqrt {2}}\right )+\log (x+1) \log \left (1-\frac {x+1}{1-\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\text {Li}_2\left (\frac {x+1}{1+\sqrt {2}}\right )+\log (x+1) \log \left (1-\frac {x+1}{1+\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\left (\log \left (\sqrt {2}-x\right )-\log \left (x+\sqrt {2}\right )\right ) \left (\log \left (1-x^2\right )-\log (x-1)-\log (x+1)\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\log \left (-x^{2} + 1\right )}{x^{2} - 2}, x\right ) \]
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 \[ \int -\frac {\log \left (-x^{2} + 1\right )}{x^{2} - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 214, normalized size = 0.90 \[ -\frac {\sqrt {2}\, \ln \left (\frac {x -1}{-\sqrt {2}-1}\right ) \ln \left (x +\sqrt {2}\right )}{4}+\frac {\sqrt {2}\, \ln \left (\frac {x -1}{\sqrt {2}-1}\right ) \ln \left (x -\sqrt {2}\right )}{4}+\frac {\sqrt {2}\, \ln \left (\frac {x +1}{1+\sqrt {2}}\right ) \ln \left (x -\sqrt {2}\right )}{4}-\frac {\sqrt {2}\, \ln \left (\frac {x +1}{-\sqrt {2}+1}\right ) \ln \left (x +\sqrt {2}\right )}{4}-\frac {\sqrt {2}\, \ln \left (x -\sqrt {2}\right ) \ln \left (-x^{2}+1\right )}{4}+\frac {\sqrt {2}\, \ln \left (x +\sqrt {2}\right ) \ln \left (-x^{2}+1\right )}{4}-\frac {\sqrt {2}\, \dilog \left (\frac {x -1}{-\sqrt {2}-1}\right )}{4}+\frac {\sqrt {2}\, \dilog \left (\frac {x -1}{\sqrt {2}-1}\right )}{4}+\frac {\sqrt {2}\, \dilog \left (\frac {x +1}{1+\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \dilog \left (\frac {x +1}{-\sqrt {2}+1}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 208, normalized size = 0.87 \[ \frac {1}{4} \, \sqrt {2} {\left ({\left (\log \left (2 \, x + 2 \, \sqrt {2}\right ) - \log \left (2 \, x - 2 \, \sqrt {2}\right )\right )} \log \left (-x^{2} + 1\right ) - \log \left (x + \sqrt {2}\right ) \log \left (-\frac {x + \sqrt {2}}{\sqrt {2} + 1} + 1\right ) + \log \left (x - \sqrt {2}\right ) \log \left (\frac {x - \sqrt {2}}{\sqrt {2} + 1} + 1\right ) - \log \left (x + \sqrt {2}\right ) \log \left (-\frac {x + \sqrt {2}}{\sqrt {2} - 1} + 1\right ) + \log \left (x - \sqrt {2}\right ) \log \left (\frac {x - \sqrt {2}}{\sqrt {2} - 1} + 1\right ) - {\rm Li}_2\left (\frac {x + \sqrt {2}}{\sqrt {2} + 1}\right ) + {\rm Li}_2\left (-\frac {x - \sqrt {2}}{\sqrt {2} + 1}\right ) - {\rm Li}_2\left (\frac {x + \sqrt {2}}{\sqrt {2} - 1}\right ) + {\rm Li}_2\left (-\frac {x - \sqrt {2}}{\sqrt {2} - 1}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {\ln \left (1-x^2\right )}{x^2-2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\log {\left (1 - x^{2} \right )}}{x^{2} - 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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