Optimal. Leaf size=299 \[ \frac {\left (1+2 a^4 b^4\right ) x \sqrt {-b^4+a^4 x^4}}{4 a^4 b^4 \left (b^4-a^4 x^4\right )}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (-1+2 a^4 b^4\right ) \text {ArcTan}\left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a^5 b^5}+\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \left (-1+2 a^4 b^4\right ) \tanh ^{-1}\left (\frac {(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt {-b^4+a^4 x^4}}{2 \sqrt {3-2 \sqrt {2}} a b x}\right )}{a^5 b^5}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \left (-1+2 a^4 b^4\right ) \tanh ^{-1}\left (\frac {(1+i) b^2+(1-i) a^2 x^2+(1-i) \sqrt {-b^4+a^4 x^4}}{2 \sqrt {3+2 \sqrt {2}} a b x}\right )}{a^5 b^5} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.56, antiderivative size = 338, normalized size of antiderivative = 1.13, number of steps
used = 20, number of rules used = 11, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.234, Rules used = {6857, 230,
227, 1469, 541, 537, 418, 1225, 1713, 209, 212} \begin {gather*} \frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\text {ArcSin}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {a^4 x^4-b^4}}-\frac {\left (1-2 a^4 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \left (-a^4\right )^{5/4} b^5}-\frac {x \left (\frac {1}{a^4 b^4}+2\right )}{4 \sqrt {a^4 x^4-b^4}}-\frac {\left (1-2 a^4 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{8 \sqrt {2} \left (-a^4\right )^{5/4} b^5}+\frac {\left (1-2 a^4 b^4\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\text {ArcSin}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a^5 b^3 \sqrt {a^4 x^4-b^4}}-\frac {\left (2 a^4 b^4+1\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\text {ArcSin}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a^5 b^3 \sqrt {a^4 x^4-b^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 227
Rule 230
Rule 418
Rule 537
Rule 541
Rule 1225
Rule 1469
Rule 1713
Rule 6857
Rubi steps
\begin {align*} \int \frac {b^8+x^4+a^8 x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx &=\int \left (\frac {1}{\sqrt {-b^4+a^4 x^4}}+\frac {2 b^8+x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx+\int \frac {2 b^8+x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx\\ &=\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{\sqrt {-b^4+a^4 x^4}}+\int \frac {2 b^8+x^4}{\left (-b^4+a^4 x^4\right )^{3/2} \left (b^4+a^4 x^4\right )} \, dx\\ &=-\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {\int \frac {b^8 \left (1-6 a^4 b^4\right )-a^4 b^4 \left (1+2 a^4 b^4\right ) x^4}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx}{4 a^4 b^8}\\ &=-\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} \left (2+\frac {1}{a^4 b^4}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{2} \left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx\\ &=-\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{4 b^4}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{4 b^4}-\frac {\left (\left (2+\frac {1}{a^4 b^4}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {\left (2+\frac {1}{a^4 b^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}+2 \frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{8 b^4}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1-\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 b^4}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \int \frac {1+\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 b^4}\\ &=-\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {\left (2+\frac {1}{a^4 b^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \text {Subst}\left (\int \frac {1}{1-2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 b^4}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \text {Subst}\left (\int \frac {1}{1+2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 b^4}+2 \frac {\left (\left (\frac {1}{a^4}-2 b^4\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 b^4 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {\left (2+\frac {1}{a^4 b^4}\right ) x}{4 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b^5}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{8 \sqrt {2} \sqrt [4]{-a^4} b^5}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {\left (2+\frac {1}{a^4 b^4}\right ) b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{a^4}-2 b^4\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a b^3 \sqrt {-b^4+a^4 x^4}}\\ \end {align*}
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Mathematica [A]
time = 1.06, size = 151, normalized size = 0.51 \begin {gather*} \frac {\left (\frac {1}{16}+\frac {i}{16}\right ) \left (-\frac {(2-2 i) a b \left (1+2 a^4 b^4\right ) x}{\sqrt {-b^4+a^4 x^4}}+\left (-1+2 a^4 b^4\right ) \text {ArcTan}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b^4+a^4 x^4}}{a b x}\right )+2 i \left (-1+2 a^4 b^4\right ) \text {ArcTan}\left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )\right )}{a^5 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.18, size = 1031, normalized size = 3.45
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{16 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{32 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{16 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{16 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, x}{2 \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {2}\, x}{4 a^{4} b^{4} \sqrt {a^{4} x^{4}-b^{4}}}\right ) \sqrt {2}}{2}\) | \(515\) |
default | \(\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\left (-2 a^{4} b^{4}-1\right ) \left (\frac {a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x +\frac {b}{a}\right ) \left (a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}\right )}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{8 a^{4} b^{3}}+\frac {\left (-2 a^{4} b^{4}+1\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}+b^{4}\right )}{\sum }\frac {-\frac {\sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) a^{4}}{\sqrt {-2 b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {-b^{4}}}+\frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{4} \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}}{b^{2}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{4} \sqrt {a^{4} x^{4}-b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 a^{8}}+\frac {\left (2 a^{4} b^{4}+1\right ) \left (-\frac {a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x -\frac {b}{a}\right ) \left (a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}\right )}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{8 b^{3} a^{4}}+\frac {\left (-2 a^{4} b^{4}-1\right ) \left (-\frac {\left (a^{4} x^{2}-a^{2} b^{2}\right ) x}{2 b^{4} a^{2} \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) \left (a^{4} x^{2}-a^{2} b^{2}\right )}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{4 a^{4} b^{2}}\) | \(1031\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.28, size = 215, normalized size = 0.72 \begin {gather*} -\frac {4 \, {\left (2 \, a^{5} b^{5} + a b\right )} \sqrt {a^{4} x^{4} - b^{4}} x + 2 \, {\left (2 \, a^{4} b^{8} - {\left (2 \, a^{8} b^{4} - a^{4}\right )} x^{4} - b^{4}\right )} \arctan \left (\frac {\sqrt {a^{4} x^{4} - b^{4}} a x}{a^{2} b x^{2} + b^{3}}\right ) + {\left (2 \, a^{4} b^{8} - {\left (2 \, a^{8} b^{4} - a^{4}\right )} x^{4} - b^{4}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} - b^{4} - 2 \, \sqrt {a^{4} x^{4} - b^{4}} a b x}{a^{4} x^{4} + b^{4}}\right )}{16 \, {\left (a^{9} b^{5} x^{4} - a^{5} b^{9}\right )}} \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 {a^{8} x^{8} + b^{8} + x^{4}}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} + b^{4}\right )}\, 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 = -0.00 \begin {gather*} \int -\frac {a^8\,x^8+b^8+x^4}{\sqrt {a^4\,x^4-b^4}\,\left (b^8-a^8\,x^8\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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