Optimal. Leaf size=78 \[ \sqrt {2 \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\sqrt {2+\sqrt {5}} \left (\sqrt {x^2+1}+x\right )\right )-\sqrt {2 \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\sqrt {5}-2} \left (\sqrt {x^2+1}+x\right )\right ) \]
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Rubi [B] time = 0.57, antiderivative size = 319, normalized size of antiderivative = 4.09, number of steps used = 25, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {6742, 261, 1130, 203, 207, 1251, 824, 707, 1093, 1247, 699, 1279} \[ -\sqrt {\frac {2}{5} \left (\sqrt {5}-1\right )} \tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {x^2+1}\right )-\sqrt {\frac {2}{5 \left (\sqrt {5}-1\right )}} \tan ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {x^2+1}\right )+\sqrt {\frac {2}{5} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x^2+1}\right )-\sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x^2+1}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-2 \sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )-2 \sqrt {\frac {2}{5 \left (\sqrt {5}-1\right )}} \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 207
Rule 261
Rule 699
Rule 707
Rule 824
Rule 1093
Rule 1130
Rule 1247
Rule 1251
Rule 1279
Rule 6742
Rubi steps
\begin {align*} \int -\frac {x+2 \sqrt {1+x^2}}{x+x^3+\sqrt {1+x^2}} \, dx &=-\int \left (\frac {x}{x+x^3+\sqrt {1+x^2}}+\frac {2 \sqrt {1+x^2}}{x+x^3+\sqrt {1+x^2}}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^2}}{x+x^3+\sqrt {1+x^2}} \, dx\right )-\int \frac {x}{x+x^3+\sqrt {1+x^2}} \, dx\\ &=-\left (2 \int \left (1+\frac {x \sqrt {1+x^2}}{-1+x^2+x^4}-\frac {x^2 \left (1+x^2\right )}{-1+x^2+x^4}\right ) \, dx\right )-\int \left (\frac {x}{\sqrt {1+x^2}}+\frac {x^2}{-1+x^2+x^4}-\frac {x^3 \sqrt {1+x^2}}{-1+x^2+x^4}\right ) \, dx\\ &=-2 x-2 \int \frac {x \sqrt {1+x^2}}{-1+x^2+x^4} \, dx+2 \int \frac {x^2 \left (1+x^2\right )}{-1+x^2+x^4} \, dx-\int \frac {x}{\sqrt {1+x^2}} \, dx-\int \frac {x^2}{-1+x^2+x^4} \, dx+\int \frac {x^3 \sqrt {1+x^2}}{-1+x^2+x^4} \, dx\\ &=-\sqrt {1+x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x \sqrt {1+x}}{-1+x+x^2} \, dx,x,x^2\right )+2 \int \frac {1}{-1+x^2+x^4} \, dx+\frac {1}{10} \left (-5+\sqrt {5}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx-\frac {1}{10} \left (5+\sqrt {5}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx-\operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{-1+x+x^2} \, dx,x,x^2\right )\\ &=-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x} \left (-1+x+x^2\right )} \, dx,x,x^2\right )-2 \operatorname {Subst}\left (\int \frac {x^2}{-1-x^2+x^4} \, dx,x,\sqrt {1+x^2}\right )+\frac {2 \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx}{\sqrt {5}}-\frac {2 \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx}{\sqrt {5}}\\ &=-2 \sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-2 \sqrt {\frac {2}{5 \left (-1+\sqrt {5}\right )}} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\frac {1}{5} \left (5-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+x^2}\right )-\frac {1}{5} \left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+x^2}\right )+\operatorname {Subst}\left (\int \frac {1}{-1-x^2+x^4} \, dx,x,\sqrt {1+x^2}\right )\\ &=-2 \sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {2}{5} \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt {1+x^2}\right )-2 \sqrt {\frac {2}{5 \left (-1+\sqrt {5}\right )}} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+\sqrt {\frac {2}{5} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {1+x^2}\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+x^2}\right )}{\sqrt {5}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+x^2}\right )}{\sqrt {5}}\\ &=-2 \sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {2}{5 \left (-1+\sqrt {5}\right )}} \tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt {1+x^2}\right )-\sqrt {\frac {2}{5} \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} \sqrt {1+x^2}\right )-2 \sqrt {\frac {2}{5 \left (-1+\sqrt {5}\right )}} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {1+x^2}\right )+\sqrt {\frac {2}{5} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {1+x^2}\right )\\ \end {align*}
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Mathematica [F] time = 0.43, size = 34, normalized size = 0.44 \[ -\int \frac {2 \sqrt {x^2+1}+x}{x^3+\sqrt {x^2+1}+x} \, dx \]
Antiderivative was successfully verified.
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fricas [B] time = 1.43, size = 383, normalized size = 4.91 \[ \sqrt {2} \sqrt {\sqrt {5} + 1} \arctan \left (\frac {1}{4} \, \sqrt {2} \sqrt {4 \, x^{4} + 4 \, x^{2} + \sqrt {5} {\left (2 \, x^{2} + 1\right )} - 2 \, {\left (2 \, x^{3} + \sqrt {5} x + x\right )} \sqrt {x^{2} + 1} + 1} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + 1}\right )} \sqrt {\sqrt {5} + 1} - \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} + 1} \sqrt {\sqrt {5} + 1}\right ) + \sqrt {2} \sqrt {\sqrt {5} + 1} \arctan \left (\frac {1}{8} \, \sqrt {4 \, x^{2} + 2 \, \sqrt {5} + 2} {\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 1} - \frac {1}{4} \, {\left (\sqrt {5} \sqrt {2} x - \sqrt {2} x\right )} \sqrt {\sqrt {5} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (4 \, x^{2} - 4 \, \sqrt {x^{2} + 1} x + {\left (\sqrt {5} \sqrt {2} x - \sqrt {x^{2} + 1} {\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} + \sqrt {2} x\right )} \sqrt {\sqrt {5} - 1} + 4\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (4 \, x^{2} - 4 \, \sqrt {x^{2} + 1} x - {\left (\sqrt {5} \sqrt {2} x - \sqrt {x^{2} + 1} {\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} + \sqrt {2} x\right )} \sqrt {\sqrt {5} - 1} + 4\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (2 \, x + \sqrt {2} \sqrt {\sqrt {5} - 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (2 \, x - \sqrt {2} \sqrt {\sqrt {5} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.86, size = 218, normalized size = 2.79 \[ -\frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (-\frac {x - \sqrt {x^{2} + 1} + \frac {1}{x - \sqrt {x^{2} + 1}}}{\sqrt {2 \, \sqrt {5} - 2}}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (-x + \sqrt {x^{2} + 1} + \sqrt {2 \, \sqrt {5} + 2} - \frac {1}{x - \sqrt {x^{2} + 1}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | -x + \sqrt {x^{2} + 1} - \sqrt {2 \, \sqrt {5} + 2} - \frac {1}{x - \sqrt {x^{2} + 1}} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 438, normalized size = 5.62 \[ -\frac {x}{2}+\frac {3 \sqrt {5}\, \arctanh \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {-2+\sqrt {5}}}\right )}{10 \sqrt {-2+\sqrt {5}}}-\frac {\arctanh \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {-2+\sqrt {5}}}\right )}{2 \sqrt {-2+\sqrt {5}}}+\frac {2 \sqrt {5}\, \sqrt {-2+\sqrt {5}}\, \arctanh \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {-2+\sqrt {5}}}\right )}{5}+\frac {\arctanh \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {2+\sqrt {5}}}\right )}{2 \sqrt {2+\sqrt {5}}}+\frac {\sqrt {5}\, \arctanh \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {2+\sqrt {5}}}\right )}{2 \sqrt {2+\sqrt {5}}}-\frac {\sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}+\frac {\arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}-\frac {\arctan \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {-2+\sqrt {5}}}\right )}{2 \sqrt {-2+\sqrt {5}}}+\frac {\sqrt {5}\, \arctan \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {-2+\sqrt {5}}}\right )}{2 \sqrt {-2+\sqrt {5}}}+\frac {3 \sqrt {5}\, \arctan \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {2+\sqrt {5}}}\right )}{10 \sqrt {2+\sqrt {5}}}+\frac {\arctan \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {2+\sqrt {5}}}\right )}{2 \sqrt {2+\sqrt {5}}}-\frac {2 \sqrt {2+\sqrt {5}}\, \sqrt {5}\, \arctan \left (\frac {-x +\sqrt {x^{2}+1}}{\sqrt {2+\sqrt {5}}}\right )}{5}-\frac {\sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}}-\frac {\arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}}-\frac {\sqrt {x^{2}+1}}{2}+\frac {1}{-2 x +2 \sqrt {x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -x - \frac {1}{2} \, \arctan \relax (x) + \int \frac {2 \, x^{6} + 3 \, x^{4} - x^{2} - 1}{2 \, {\left (x^{6} + 2 \, x^{4} + 2 \, x^{2} + 2 \, {\left (x^{3} + x\right )} \sqrt {x^{2} + 1} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.38, size = 649, normalized size = 8.32 \[ \frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}-\frac {5}{2}\right )}{2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}-\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}-\frac {5}{2}\right )}{2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}+\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {5}{2}\right )}{2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}-\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {5}{2}\right )}{2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}-\frac {\left (\ln \left (x-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-1}}{2}\right )-\ln \left (\frac {\sqrt {2}\,x\,\sqrt {\sqrt {5}-1}}{2}+\frac {\sqrt {2}\,\sqrt {x^2+1}\,\sqrt {\sqrt {5}+1}}{2}+1\right )\right )\,\left (\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+2\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}}-\frac {\left (\ln \left (x+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-1}}{2}\right )-\ln \left (\frac {\sqrt {2}\,\sqrt {x^2+1}\,\sqrt {\sqrt {5}+1}}{2}-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {5}-1}}{2}+1\right )\right )\,\left (\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+2\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}}+\frac {\left (\ln \left (\frac {\sqrt {2}\,\sqrt {x^2+1}\,\sqrt {1-\sqrt {5}}}{2}-\frac {\sqrt {2}\,x\,\sqrt {-\sqrt {5}-1}}{2}+1\right )-\ln \left (x+\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\right )\,\left (\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+2\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}+\frac {\left (\ln \left (\frac {\sqrt {2}\,x\,\sqrt {-\sqrt {5}-1}}{2}+\frac {\sqrt {2}\,\sqrt {x^2+1}\,\sqrt {1-\sqrt {5}}}{2}+1\right )-\ln \left (x-\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\right )\,\left (\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+2\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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