Optimal. Leaf size=150 \[ \frac {2}{3} \sqrt [3]{2} \log \left (\sqrt [3]{2} \sqrt [3]{2 x^3-1}-2 x\right )-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt [3]{2} \sqrt [3]{2 x^3-1}+x}\right )}{\sqrt {3}}+\frac {\left (2 x^3-1\right )^{2/3} \left (9 x^3-2\right )}{10 x^5}-\frac {1}{3} \sqrt [3]{2} \log \left (2 \sqrt [3]{2} \sqrt [3]{2 x^3-1} x+2^{2/3} \left (2 x^3-1\right )^{2/3}+4 x^2\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 159, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {2}{3} \sqrt [3]{2} \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{2 x^3-1}}\right )-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\frac {2\ 2^{2/3} x}{\sqrt [3]{2 x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (2 x^3-1\right )^{2/3}}{5 x^5}+\frac {9 \left (2 x^3-1\right )^{2/3}}{10 x^2}-\frac {1}{3} \sqrt [3]{2} \log \left (\frac {2^{2/3} x}{\sqrt [3]{2 x^3-1}}+\frac {2 \sqrt [3]{2} x^2}{\left (2 x^3-1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 200
Rule 204
Rule 377
Rule 580
Rule 583
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right ) \left (-1+2 x^3\right )^{2/3}}{x^6 \left (1+2 x^3\right )} \, dx &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {1}{5} \int \frac {9-2 x^3}{x^3 \sqrt [3]{-1+2 x^3} \left (1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}+\frac {1}{10} \int -\frac {40}{\sqrt [3]{-1+2 x^3} \left (1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-4 \int \frac {1}{\sqrt [3]{-1+2 x^3} \left (1+2 x^3\right )} \, dx\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-4 \operatorname {Subst}\left (\int \frac {1}{1-4 x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1-2^{2/3} x} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {2+2^{2/3} x}{1+2^{2/3} x+2 \sqrt [3]{2} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}+\frac {2}{3} \sqrt [3]{2} \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )-2 \operatorname {Subst}\left (\int \frac {1}{1+2^{2/3} x+2 \sqrt [3]{2} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{3} \sqrt [3]{2} \operatorname {Subst}\left (\int \frac {2^{2/3}+4 \sqrt [3]{2} x}{1+2^{2/3} x+2 \sqrt [3]{2} x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}+\frac {2}{3} \sqrt [3]{2} \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {2 \sqrt [3]{2} x^2}{\left (-1+2 x^3\right )^{2/3}}+\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )+\left (2 \sqrt [3]{2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2\ 2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )\\ &=-\frac {\left (-1+2 x^3\right )^{2/3}}{5 x^5}+\frac {9 \left (-1+2 x^3\right )^{2/3}}{10 x^2}-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {1+\frac {2\ 2^{2/3} x}{\sqrt [3]{-1+2 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \sqrt [3]{2} \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (1+\frac {2 \sqrt [3]{2} x^2}{\left (-1+2 x^3\right )^{2/3}}+\frac {2^{2/3} x}{\sqrt [3]{-1+2 x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.17, size = 139, normalized size = 0.93 \begin {gather*} \left (\frac {9}{10 x^2}-\frac {1}{5 x^5}\right ) \left (2 x^3-1\right )^{2/3}-\frac {1}{3} \sqrt [3]{2} \left (-2 \log \left (1-\frac {2^{2/3} x}{\sqrt [3]{2-x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2\ 2^{2/3} x}{\sqrt [3]{2-x^3}}+1}{\sqrt {3}}\right )+\log \left (\frac {2^{2/3} x}{\sqrt [3]{2-x^3}}+\frac {2 \sqrt [3]{2} x^2}{\left (2-x^3\right )^{2/3}}+1\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.37, size = 150, normalized size = 1.00 \begin {gather*} \frac {\left (-1+2 x^3\right )^{2/3} \left (-2+9 x^3\right )}{10 x^5}-\frac {2 \sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+\sqrt [3]{2} \sqrt [3]{-1+2 x^3}}\right )}{\sqrt {3}}+\frac {2}{3} \sqrt [3]{2} \log \left (-2 x+\sqrt [3]{2} \sqrt [3]{-1+2 x^3}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (4 x^2+2 \sqrt [3]{2} x \sqrt [3]{-1+2 x^3}+2^{2/3} \left (-1+2 x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.84, size = 290, normalized size = 1.93 \begin {gather*} -\frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (20 \, x^{7} + 8 \, x^{4} - x\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {3} 2^{\frac {1}{3}} {\left (76 \, x^{8} - 32 \, x^{5} + x^{2}\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (568 \, x^{9} - 444 \, x^{6} + 66 \, x^{3} - 1\right )}}{3 \, {\left (872 \, x^{9} - 420 \, x^{6} + 6 \, x^{3} + 1\right )}}\right ) - 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 2^{\frac {2}{3}} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 2^{\frac {1}{3}} {\left (2 \, x^{3} + 1\right )}}{2 \, x^{3} + 1}\right ) + 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (10 \, x^{4} - x\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (76 \, x^{6} - 32 \, x^{3} + 1\right )} + 24 \, {\left (4 \, x^{5} - x^{2}\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{4 \, x^{6} + 4 \, x^{3} + 1}\right ) - 9 \, {\left (9 \, x^{3} - 2\right )} {\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \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*} \int \frac {{\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}}{{\left (2 \, x^{3} + 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 17.26, size = 631, normalized size = 4.21
method | result | size |
risch | \(\frac {18 x^{6}-13 x^{3}+2}{10 x^{5} \left (2 x^{3}-1\right )^{\frac {1}{3}}}+\frac {2 \RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (-\frac {12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+108 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-12 \left (2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +4 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (\textit {\_Z}^{3}-2\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+54 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-5 \left (2 x^{3}-1\right )^{\frac {2}{3}} x -\RootOf \left (\textit {\_Z}^{3}-2\right )-9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{2 x^{3}+1}\right )}{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+144 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-24 \left (2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +8 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+60 \RootOf \left (\textit {\_Z}^{3}-2\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{2}-6 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}+2 \left (2 x^{3}-1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{3}-2\right )+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{2 x^{3}+1}\right )\) | \(631\) |
trager | \(\frac {\left (2 x^{3}-1\right )^{\frac {2}{3}} \left (9 x^{3}-2\right )}{10 x^{5}}+\frac {2 \RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (\frac {204 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+396 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-720 \left (2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +240 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+1422 \RootOf \left (\textit {\_Z}^{3}-2\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{2}-204 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3}-396 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2}-374 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-726 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-3 \left (2 x^{3}-1\right )^{\frac {2}{3}} x +17 \RootOf \left (\textit {\_Z}^{3}-2\right )+33 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{2 x^{3}+1}\right )}{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {2676 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+96912 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-64656 \left (2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +21552 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+24372 \RootOf \left (\textit {\_Z}^{3}-2\right ) \left (2 x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{2}-2676 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3}-96912 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+5798 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+209976 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-17490 \left (2 x^{3}-1\right )^{\frac {2}{3}} x -1115 \RootOf \left (\textit {\_Z}^{3}-2\right )-40380 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )}{2 x^{3}+1}\right )\) | \(765\) |
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 \begin {gather*} \int \frac {{\left (2 \, x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} + 1\right )}}{{\left (2 \, x^{3} + 1\right )} x^{6}}\,{d x} \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 {\left (x^3+1\right )\,{\left (2\,x^3-1\right )}^{2/3}}{x^6\,\left (2\,x^3+1\right )} \,d x \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 {\left (x + 1\right ) \left (2 x^{3} - 1\right )^{\frac {2}{3}} \left (x^{2} - x + 1\right )}{x^{6} \left (2 x^{3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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