Optimal. Leaf size=88 \[ -\frac {\tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1-\sqrt {2} x\right )}{\sqrt {2}}-\frac {\text {PolyLog}\left (2,-\frac {\sqrt {2}-2 x}{2-\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {\text {PolyLog}\left (2,\frac {\sqrt {2}-2 x}{2+\sqrt {2}}\right )}{2 \sqrt {2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 108, normalized size of antiderivative = 1.23, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6057, 2449,
2352, 2497} \begin {gather*} -\frac {\text {Li}_2\left (1-\frac {2}{x+1}\right )}{2 \sqrt {2}}+\frac {\text {Li}_2\left (\frac {2 \left (1+\sqrt {2}\right ) \left (1-\sqrt {2} x\right )}{x+1}+1\right )}{2 \sqrt {2}}+\frac {\log \left (\frac {2}{x+1}\right ) \tanh ^{-1}(x)}{\sqrt {2}}-\frac {\log \left (-\frac {2 \left (1+\sqrt {2}\right ) \left (1-\sqrt {2} x\right )}{x+1}\right ) \tanh ^{-1}(x)}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 2497
Rule 6057
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(x)}{1-\sqrt {2} x} \, dx &=\frac {\tanh ^{-1}(x) \log \left (\frac {2}{1+x}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}(x) \log \left (-\frac {2 \left (1+\sqrt {2}\right ) \left (1-\sqrt {2} x\right )}{1+x}\right )}{\sqrt {2}}-\frac {\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx}{\sqrt {2}}+\frac {\int \frac {\log \left (\frac {2 \left (1-\sqrt {2} x\right )}{\left (1-\sqrt {2}\right ) (1+x)}\right )}{1-x^2} \, dx}{\sqrt {2}}\\ &=\frac {\tanh ^{-1}(x) \log \left (\frac {2}{1+x}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}(x) \log \left (-\frac {2 \left (1+\sqrt {2}\right ) \left (1-\sqrt {2} x\right )}{1+x}\right )}{\sqrt {2}}+\frac {\text {Li}_2\left (1+\frac {2 \left (1+\sqrt {2}\right ) \left (1-\sqrt {2} x\right )}{1+x}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+x}\right )}{\sqrt {2}}\\ &=\frac {\tanh ^{-1}(x) \log \left (\frac {2}{1+x}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}(x) \log \left (-\frac {2 \left (1+\sqrt {2}\right ) \left (1-\sqrt {2} x\right )}{1+x}\right )}{\sqrt {2}}-\frac {\text {Li}_2\left (1-\frac {2}{1+x}\right )}{2 \sqrt {2}}+\frac {\text {Li}_2\left (1+\frac {2 \left (1+\sqrt {2}\right ) \left (1-\sqrt {2} x\right )}{1+x}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.07, size = 272, normalized size = 3.09 \begin {gather*} \frac {\pi ^2-4 \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right )^2-4 i \pi \tanh ^{-1}(x)+8 \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \tanh ^{-1}(x)-8 \tanh ^{-1}(x)^2+8 \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1-e^{2 \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right )-2 \tanh ^{-1}(x)}\right )-8 \tanh ^{-1}(x) \log \left (1-e^{2 \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right )-2 \tanh ^{-1}(x)}\right )+4 i \pi \log \left (1+e^{2 \tanh ^{-1}(x)}\right )+8 \tanh ^{-1}(x) \log \left (1+e^{2 \tanh ^{-1}(x)}\right )-4 i \pi \log \left (\frac {2}{\sqrt {1-x^2}}\right )-8 \tanh ^{-1}(x) \log \left (\frac {2}{\sqrt {1-x^2}}\right )-4 \tanh ^{-1}(x) \log \left (1-x^2\right )-8 \tanh ^{-1}(x) \log \left (-i \sinh \left (\tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right )-\tanh ^{-1}(x)\right )\right )-8 \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (-2 i \sinh \left (\tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right )-\tanh ^{-1}(x)\right )\right )+8 \tanh ^{-1}(x) \log \left (-2 i \sinh \left (\tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right )-\tanh ^{-1}(x)\right )\right )+4 \text {PolyLog}\left (2,e^{2 \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right )-2 \tanh ^{-1}(x)}\right )+4 \text {PolyLog}\left (2,-e^{2 \tanh ^{-1}(x)}\right )}{8 \sqrt {2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.18, size = 127, normalized size = 1.44
method | result | size |
default | \(-\frac {\ln \left (x \sqrt {2}-1\right ) \sqrt {2}\, \arctanh \left (x \right )}{2}-\frac {\sqrt {2}\, \ln \left (x \sqrt {2}-1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{\sqrt {2}-1}\right )}{4}+\frac {\sqrt {2}\, \ln \left (x \sqrt {2}-1\right ) \ln \left (\frac {\sqrt {2}+x \sqrt {2}}{1+\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \dilog \left (\frac {\sqrt {2}-x \sqrt {2}}{\sqrt {2}-1}\right )}{4}+\frac {\sqrt {2}\, \dilog \left (\frac {\sqrt {2}+x \sqrt {2}}{1+\sqrt {2}}\right )}{4}\) | \(127\) |
risch | \(\frac {\sqrt {2}\, \ln \left (\frac {-2 x +\sqrt {2}}{2+\sqrt {2}}\right ) \ln \left (\frac {2+2 x}{2+\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \ln \left (\frac {-2 x +\sqrt {2}}{2+\sqrt {2}}\right ) \ln \left (1+x \right )}{4}+\frac {\sqrt {2}\, \dilog \left (\frac {2+2 x}{2+\sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \ln \left (\frac {2 x -\sqrt {2}}{2-\sqrt {2}}\right ) \ln \left (1-x \right )}{4}-\frac {\sqrt {2}\, \ln \left (\frac {2 x -\sqrt {2}}{2-\sqrt {2}}\right ) \ln \left (\frac {2-2 x}{2-\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \dilog \left (\frac {2-2 x}{2-\sqrt {2}}\right )}{4}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs.
\(2 (69) = 138\).
time = 0.46, size = 144, normalized size = 1.64 \begin {gather*} \frac {1}{4} \, \sqrt {2} {\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )} \log \left (\sqrt {2} x - 1\right ) - \frac {1}{2} \, \sqrt {2} \operatorname {artanh}\left (x\right ) \log \left (\sqrt {2} x - 1\right ) - \frac {1}{4} \, \sqrt {2} {\left (\log \left (x + 1\right ) \log \left (-\frac {\sqrt {2} x + \sqrt {2}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} x + \sqrt {2}}{\sqrt {2} + 1}\right )\right )} + \frac {1}{4} \, \sqrt {2} {\left (\log \left (x - 1\right ) \log \left (\frac {\sqrt {2} x - \sqrt {2}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (-\frac {\sqrt {2} x - \sqrt {2}}{\sqrt {2} - 1}\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 {\operatorname {atanh}{\left (x \right )}}{\sqrt {2} x - 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 {\mathrm {atanh}\left (x\right )}{\sqrt {2}\,x-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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