Optimal. Leaf size=246 \[ -\frac {11 \sqrt {\frac {1}{3} \left (1825+1089 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{1536}+\frac {11 \sqrt {\frac {1}{3} \left (1825+1089 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{1536}+\frac {25 x \left (x^2+1\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac {x \left (88 x^2+353\right )}{192 \left (x^4+2 x^2+3\right )}-\frac {11}{768} \sqrt {\frac {1}{3} \left (1089 \sqrt {3}-1825\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {11}{768} \sqrt {\frac {1}{3} \left (1089 \sqrt {3}-1825\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]
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Rubi [A] time = 0.28, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1668, 1678, 1169, 634, 618, 204, 628} \[ \frac {25 x \left (x^2+1\right )}{16 \left (x^4+2 x^2+3\right )^2}-\frac {x \left (88 x^2+353\right )}{192 \left (x^4+2 x^2+3\right )}-\frac {11 \sqrt {\frac {1}{3} \left (1825+1089 \sqrt {3}\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{1536}+\frac {11 \sqrt {\frac {1}{3} \left (1825+1089 \sqrt {3}\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{1536}-\frac {11}{768} \sqrt {\frac {1}{3} \left (1089 \sqrt {3}-1825\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {11}{768} \sqrt {\frac {1}{3} \left (1089 \sqrt {3}-1825\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1169
Rule 1668
Rule 1678
Rubi steps
\begin {align*} \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac {1}{96} \int \frac {-150+78 x^2+480 x^4}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {6072-2112 x^2}{3+2 x^2+x^4} \, dx}{4608}\\ &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}+\frac {\int \frac {6072 \sqrt {2 \left (-1+\sqrt {3}\right )}-\left (6072+2112 \sqrt {3}\right ) x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{9216 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\int \frac {6072 \sqrt {2 \left (-1+\sqrt {3}\right )}+\left (6072+2112 \sqrt {3}\right ) x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{9216 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac {\left (11 \left (24-23 \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{2304}-\frac {\left (11 \left (24-23 \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{2304}-\frac {\left (11 \left (23+8 \sqrt {3}\right )\right ) \int \frac {-\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{768 \sqrt {6 \left (-1+\sqrt {3}\right )}}+\frac {\left (11 \left (23+8 \sqrt {3}\right )\right ) \int \frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2} \, dx}{768 \sqrt {6 \left (-1+\sqrt {3}\right )}}\\ &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac {11}{768} \sqrt {\frac {1825}{12}+\frac {363 \sqrt {3}}{4}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {11}{768} \sqrt {\frac {1825}{12}+\frac {363 \sqrt {3}}{4}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {\left (11 \left (24-23 \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {3}\right )-x^2} \, dx,x,-\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x\right )}{1152}+\frac {\left (11 \left (24-23 \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {3}\right )-x^2} \, dx,x,\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x\right )}{1152}\\ &=\frac {25 x \left (1+x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}-\frac {x \left (353+88 x^2\right )}{192 \left (3+2 x^2+x^4\right )}-\frac {11}{768} \sqrt {\frac {1}{3} \left (-1825+1089 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {11}{768} \sqrt {\frac {1}{3} \left (-1825+1089 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {11}{768} \sqrt {\frac {1825}{12}+\frac {363 \sqrt {3}}{4}} \log \left (\sqrt {3}-\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )+\frac {11}{768} \sqrt {\frac {1825}{12}+\frac {363 \sqrt {3}}{4}} \log \left (\sqrt {3}+\sqrt {2 \left (-1+\sqrt {3}\right )} x+x^2\right )\\ \end {align*}
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Mathematica [C] time = 0.30, size = 133, normalized size = 0.54 \[ \frac {1}{768} \left (-\frac {4 x \left (88 x^6+529 x^4+670 x^2+759\right )}{\left (x^4+2 x^2+3\right )^2}-\frac {11 i \left (31 \sqrt {2}-16 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {11 i \left (31 \sqrt {2}+16 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 570, normalized size = 2.32 \[ -\frac {12811392 \, x^{7} + 77013936 \, x^{5} + 1348 \, \sqrt {6} 3^{\frac {3}{4}} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} \arctan \left (\frac {1}{2226179538} \, \sqrt {3707} \sqrt {6} 3^{\frac {3}{4}} \sqrt {\sqrt {6} 3^{\frac {1}{4}} {\left (8 \, \sqrt {3} x + 23 \, x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + 33363 \, x^{2} + 33363 \, \sqrt {3}} {\left (23 \, \sqrt {3} \sqrt {2} + 24 \, \sqrt {2}\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} - \frac {1}{200178} \, \sqrt {6} 3^{\frac {3}{4}} {\left (23 \, \sqrt {3} \sqrt {2} x + 24 \, \sqrt {2} x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} - \frac {1}{2} \, \sqrt {3} \sqrt {2} + \frac {1}{2} \, \sqrt {2}\right ) + 1348 \, \sqrt {6} 3^{\frac {3}{4}} \sqrt {2} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} \arctan \left (\frac {1}{2226179538} \, \sqrt {3707} \sqrt {6} 3^{\frac {3}{4}} \sqrt {-\sqrt {6} 3^{\frac {1}{4}} {\left (8 \, \sqrt {3} x + 23 \, x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + 33363 \, x^{2} + 33363 \, \sqrt {3}} {\left (23 \, \sqrt {3} \sqrt {2} + 24 \, \sqrt {2}\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} - \frac {1}{200178} \, \sqrt {6} 3^{\frac {3}{4}} {\left (23 \, \sqrt {3} \sqrt {2} x + 24 \, \sqrt {2} x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + \frac {1}{2} \, \sqrt {3} \sqrt {2} - \frac {1}{2} \, \sqrt {2}\right ) - \sqrt {6} 3^{\frac {1}{4}} {\left (3267 \, x^{8} + 13068 \, x^{6} + 32670 \, x^{4} + 39204 \, x^{2} + 1825 \, \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 29403\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} \log \left (\sqrt {6} 3^{\frac {1}{4}} {\left (8 \, \sqrt {3} x + 23 \, x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + 33363 \, x^{2} + 33363 \, \sqrt {3}\right ) + \sqrt {6} 3^{\frac {1}{4}} {\left (3267 \, x^{8} + 13068 \, x^{6} + 32670 \, x^{4} + 39204 \, x^{2} + 1825 \, \sqrt {3} {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )} + 29403\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} \log \left (-\sqrt {6} 3^{\frac {1}{4}} {\left (8 \, \sqrt {3} x + 23 \, x\right )} \sqrt {-1987425 \, \sqrt {3} + 3557763} + 33363 \, x^{2} + 33363 \, \sqrt {3}\right ) + 97541280 \, x^{3} + 110498256 \, x}{27952128 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.69, size = 577, normalized size = 2.35 \[ \frac {11}{124416} \, \sqrt {2} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 36 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 36 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 2 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 207 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 207 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x + 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {11}{124416} \, \sqrt {2} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 36 \cdot 3^{\frac {3}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} - 36 \cdot 3^{\frac {3}{4}} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} + 2 \cdot 3^{\frac {3}{4}} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 207 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} - 207 \cdot 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x - 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {11}{248832} \, \sqrt {2} {\left (36 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 36 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 207 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 207 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} + 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) - \frac {11}{248832} \, \sqrt {2} {\left (36 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (\sqrt {3} + 3\right )} \sqrt {-6 \, \sqrt {3} + 18} - 2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (-6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 2 \cdot 3^{\frac {3}{4}} {\left (6 \, \sqrt {3} + 18\right )}^{\frac {3}{2}} + 36 \cdot 3^{\frac {3}{4}} \sqrt {6 \, \sqrt {3} + 18} {\left (\sqrt {3} - 3\right )} + 207 \cdot 3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} + 207 \cdot 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} - 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) - \frac {88 \, x^{7} + 529 \, x^{5} + 670 \, x^{3} + 759 \, x}{192 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 418, normalized size = 1.70 \[ -\frac {517 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{4608 \sqrt {2+2 \sqrt {3}}}-\frac {341 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{1536 \sqrt {2+2 \sqrt {3}}}+\frac {253 \sqrt {3}\, \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{576 \sqrt {2+2 \sqrt {3}}}-\frac {517 \left (-2+2 \sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{4608 \sqrt {2+2 \sqrt {3}}}-\frac {341 \left (-2+2 \sqrt {3}\right ) \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{1536 \sqrt {2+2 \sqrt {3}}}+\frac {253 \sqrt {3}\, \arctan \left (\frac {2 x +\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{576 \sqrt {2+2 \sqrt {3}}}-\frac {517 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{9216}-\frac {341 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}-\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{3072}+\frac {517 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{9216}+\frac {341 \sqrt {-2+2 \sqrt {3}}\, \ln \left (x^{2}+\sqrt {-2+2 \sqrt {3}}\, x +\sqrt {3}\right )}{3072}+\frac {-\frac {11}{24} x^{7}-\frac {529}{192} x^{5}-\frac {335}{96} x^{3}-\frac {253}{64} x}{\left (x^{4}+2 x^{2}+3\right )^{2}} \]
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 \[ -\frac {88 \, x^{7} + 529 \, x^{5} + 670 \, x^{3} + 759 \, x}{192 \, {\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} - \frac {11}{192} \, \int \frac {8 \, x^{2} - 23}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 174, normalized size = 0.71 \[ -\frac {\frac {11\,x^7}{24}+\frac {529\,x^5}{192}+\frac {335\,x^3}{96}+\frac {253\,x}{64}}{x^8+4\,x^6+10\,x^4+12\,x^2+9}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {10950-\sqrt {2}\,2022{}\mathrm {i}}\,448547{}\mathrm {i}}{31850496\,\left (-\frac {21081709}{10616832}+\frac {\sqrt {2}\,10316581{}\mathrm {i}}{10616832}\right )}-\frac {448547\,\sqrt {2}\,x\,\sqrt {10950-\sqrt {2}\,2022{}\mathrm {i}}}{63700992\,\left (-\frac {21081709}{10616832}+\frac {\sqrt {2}\,10316581{}\mathrm {i}}{10616832}\right )}\right )\,\sqrt {10950-\sqrt {2}\,2022{}\mathrm {i}}\,11{}\mathrm {i}}{2304}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {10950+\sqrt {2}\,2022{}\mathrm {i}}\,448547{}\mathrm {i}}{31850496\,\left (\frac {21081709}{10616832}+\frac {\sqrt {2}\,10316581{}\mathrm {i}}{10616832}\right )}+\frac {448547\,\sqrt {2}\,x\,\sqrt {10950+\sqrt {2}\,2022{}\mathrm {i}}}{63700992\,\left (\frac {21081709}{10616832}+\frac {\sqrt {2}\,10316581{}\mathrm {i}}{10616832}\right )}\right )\,\sqrt {10950+\sqrt {2}\,2022{}\mathrm {i}}\,11{}\mathrm {i}}{2304} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.33, size = 1200, normalized size = 4.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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