Optimal. Leaf size=193 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )-\frac {1}{2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )+\frac {1}{4} \text {PolyLog}\left (2,1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )+\frac {1}{4} \text {PolyLog}\left (2,1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6139, 6057,
2449, 2352, 2497} \begin {gather*} -\frac {1}{2} \text {Li}_2\left (1-\frac {2 \sqrt {2}}{x+\sqrt {2}}\right )+\frac {1}{4} \text {Li}_2\left (\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}+1\right )+\frac {1}{4} \text {Li}_2\left (1-\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )+\log \left (\frac {2 \sqrt {2}}{x+\sqrt {2}}\right ) \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{2} \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 )-\frac {1}{2} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 2497
Rule 6057
Rule 6139
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{1-x^2} \, dx &=\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\\ &=-\left (\frac {1}{2} \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{-1+x} \, dx\right )-\frac {1}{2} \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{1+x} \, dx\\ &=\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {1}{2} \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 )-\frac {1}{2} \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 )-2 \frac {\int \frac {\log \left (\frac {2}{1+\frac {x}{\sqrt {2}}}\right )}{1-\frac {x^2}{2}} \, dx}{2 \sqrt {2}}+\frac {\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}{2 \sqrt {2}}+\frac {\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}{2 \sqrt {2}}\\ &=\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {1}{2} \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 )-\frac {1}{2} \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 )+\frac {1}{4} \text {Li}_2\left (1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )+\frac {1}{4} \text {Li}_2\left (1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )-2 \left (\frac {1}{2} \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {x}{\sqrt {2}}}\right )\right )\\ &=\tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {1}{2} \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 )-\frac {1}{2} \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 )-\frac {1}{2} \text {Li}_2\left (1-\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )+\frac {1}{4} \text {Li}_2\left (1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )+\frac {1}{4} \text {Li}_2\left (1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 232, normalized size = 1.20 \begin {gather*} \frac {1}{4} \left (-4 \sinh ^{-1}(1) \tanh ^{-1}(x)+4 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1+e^{-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}\right )+2 \sinh ^{-1}(1) \log \left (1+\left (-3+2 \sqrt {2}\right ) e^{-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1+\left (-3+2 \sqrt {2}\right ) e^{-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}\right )-2 \sinh ^{-1}(1) \log \left (1-\left (3+2 \sqrt {2}\right ) e^{-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-\left (3+2 \sqrt {2}\right ) e^{-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}\right )-2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}\right )+\text {PolyLog}\left (2,\left (3-2 \sqrt {2}\right ) e^{-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}\right )+\text {PolyLog}\left (2,\left (3+2 \sqrt {2}\right ) e^{-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right )}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 251, normalized size = 1.30
method | result | size |
derivativedivides | \(-\frac {\ln \left (x^{2}-1\right ) \arctanh \left (\frac {x \sqrt {2}}{2}\right )}{2}-\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (x^{2}-1\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right )}{4}+\frac {\dilog \left (\frac {\sqrt {2}-x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}+\frac {\dilog \left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (x^{2}-1\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (\frac {\sqrt {2}+x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}-\frac {\dilog \left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}-\frac {\dilog \left (\frac {\sqrt {2}+x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}\) | \(251\) |
default | \(-\frac {\ln \left (x^{2}-1\right ) \arctanh \left (\frac {x \sqrt {2}}{2}\right )}{2}-\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (x^{2}-1\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right )}{4}+\frac {\dilog \left (\frac {\sqrt {2}-x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}+\frac {\dilog \left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (x^{2}-1\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (\frac {\sqrt {2}+x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}-\frac {\dilog \left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}-\frac {\dilog \left (\frac {\sqrt {2}+x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}\) | \(251\) |
risch | \(\frac {\ln \left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right ) \ln \left (1-\frac {x \sqrt {2}}{2}\right )}{4}-\frac {\ln \left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right ) \ln \left (\frac {2-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}-\frac {\dilog \left (\frac {2-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}-\sqrt {2}}{2-\sqrt {2}}\right ) \ln \left (1-\frac {x \sqrt {2}}{2}\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}-\sqrt {2}}{2-\sqrt {2}}\right ) \ln \left (\frac {2-x \sqrt {2}}{2-\sqrt {2}}\right )}{4}-\frac {\dilog \left (\frac {2-x \sqrt {2}}{2-\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right ) \ln \left (\frac {x \sqrt {2}+2}{2+\sqrt {2}}\right )}{4}+\frac {\dilog \left (\frac {x \sqrt {2}+2}{2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {-x \sqrt {2}-\sqrt {2}}{2-\sqrt {2}}\right ) \ln \left (\frac {x \sqrt {2}}{2}+1\right )}{4}+\frac {\ln \left (\frac {-x \sqrt {2}-\sqrt {2}}{2-\sqrt {2}}\right ) \ln \left (\frac {x \sqrt {2}+2}{2-\sqrt {2}}\right )}{4}+\frac {\dilog \left (\frac {x \sqrt {2}+2}{2-\sqrt {2}}\right )}{4}\) | \(378\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 277, normalized size = 1.44 \begin {gather*} -\frac {1}{2} \, \operatorname {artanh}\left (\frac {1}{2} \, \sqrt {2} x\right ) \log \left (x^{2} - 1\right ) - \frac {1}{4} \, \log \left (x^{2} - 1\right ) \log \left (\frac {x - \sqrt {2}}{x + \sqrt {2}}\right ) + \frac {1}{8} \, \sqrt {2} {\left (\sqrt {2} \log \left (x^{2} - 1\right ) \log \left (\frac {x - \sqrt {2}}{x + \sqrt {2}}\right ) + \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 )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \int \frac {x \operatorname {atanh}{\left (\frac {\sqrt {2} x}{2} \right )}}{x^{2} - 1}\, dx \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x}{2}\right )}{x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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