Optimal. Leaf size=385 \[ c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2-3 b c^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+3 b c^2 d^3 \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {9}{2} c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+6 c^2 d^3 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-2 b c^2 d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+6 b c^2 d^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}-b^2 c^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-3 b^2 c^2 d^3 \text {Li}_2\left (\frac {2}{c x+1}-1\right )+\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (\frac {2}{1-c x}-1\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (1-c^2 x^2\right )+b^2 c^2 d^3 \log (x) \]
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Rubi [A] time = 0.79, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 20, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {5940, 5910, 5984, 5918, 2402, 2315, 5916, 5982, 266, 36, 29, 31, 5948, 5988, 5932, 2447, 5914, 6052, 6058, 6610} \[ -3 b c^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+3 b c^2 d^3 \text {PolyLog}\left (2,\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-b^2 c^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )-3 b^2 c^2 d^3 \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )+\frac {3}{2} b^2 c^2 d^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^2 d^3 \text {PolyLog}\left (3,\frac {2}{1-c x}-1\right )+\frac {9}{2} c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+6 c^2 d^3 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-2 b c^2 d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+6 b c^2 d^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac {1}{2} b^2 c^2 d^3 \log \left (1-c^2 x^2\right )+b^2 c^2 d^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 2315
Rule 2402
Rule 2447
Rule 5910
Rule 5914
Rule 5916
Rule 5918
Rule 5932
Rule 5940
Rule 5948
Rule 5982
Rule 5984
Rule 5988
Rule 6052
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx &=\int \left (c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3}+\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2}+\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d^3 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx+\left (3 c d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\left (3 c^2 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx+\left (c^3 d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+6 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\left (b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (6 b c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx-\left (12 b c^3 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b c^4 d^3\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=4 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+6 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\left (b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (6 b c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx-\left (2 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (6 b c^3 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (6 b c^3 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {9}{2} c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+6 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-2 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+6 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )+\left (b^2 c^2 d^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (2 b^2 c^3 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx+\left (3 b^2 c^3 d^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (3 b^2 c^3 d^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (6 b^2 c^3 d^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {9}{2} c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+6 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-2 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+6 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-3 b^2 c^2 d^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )-\left (2 b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )\\ &=-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {9}{2} c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+6 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )-2 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )+6 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b^2 c^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-3 b^2 c^2 d^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (b^2 c^4 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {9}{2} c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+6 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+b^2 c^2 d^3 \log (x)-2 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (1-c^2 x^2\right )+6 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b^2 c^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )-3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-3 b^2 c^2 d^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )\\ \end {align*}
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Mathematica [C] time = 1.02, size = 461, normalized size = 1.20 \[ \frac {1}{2} d^3 \left (2 a^2 c^3 x+6 a^2 c^2 \log (x)-\frac {6 a^2 c}{x}-\frac {a^2}{x^2}-6 a b c^2 (\text {Li}_2(-c x)-\text {Li}_2(c x))+2 a b c^2 \left (\log \left (1-c^2 x^2\right )+2 c x \tanh ^{-1}(c x)\right )-\frac {6 a b c \left (c x \left (\log \left (1-c^2 x^2\right )-2 \log (c x)\right )+2 \tanh ^{-1}(c x)\right )}{x}-\frac {a b \left (c x (c x \log (1-c x)-c x \log (c x+1)+2)+2 \tanh ^{-1}(c x)\right )}{x^2}+2 b^2 c^2 \left (\text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \left ((c x-1) \tanh ^{-1}(c x)-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )\right )+6 b^2 c^2 \left (\tanh ^{-1}(c x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \text {Li}_2\left (e^{2 \tanh ^{-1}(c x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )-\frac {1}{2} \text {Li}_3\left (e^{2 \tanh ^{-1}(c x)}\right )-\frac {2}{3} \tanh ^{-1}(c x)^3-\tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+\tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+\frac {i \pi ^3}{24}\right )+\frac {b^2 \left (2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+\left (c^2 x^2-1\right ) \tanh ^{-1}(c x)^2-2 c x \tanh ^{-1}(c x)\right )}{x^2}+\frac {6 b^2 c \left (\tanh ^{-1}(c x) \left ((c x-1) \tanh ^{-1}(c x)+2 c x \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )-c x \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )\right )}{x}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c^{3} d^{3} x^{3} + 3 \, a^{2} c^{2} d^{3} x^{2} + 3 \, a^{2} c d^{3} x + a^{2} d^{3} + {\left (b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} + 3 \, b^{2} c d^{3} x + b^{2} d^{3}\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c^{3} d^{3} x^{3} + 3 \, a b c^{2} d^{3} x^{2} + 3 \, a b c d^{3} x + a b d^{3}\right )} \operatorname {artanh}\left (c x\right )}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.76, size = 1358, normalized size = 3.53 \[ \text {result too large to display} \]
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 \[ a^{2} c^{3} d^{3} x + {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b c^{2} d^{3} + 3 \, a^{2} c^{2} d^{3} \log \relax (x) - 3 \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} a b c d^{3} + \frac {1}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} a b d^{3} - \frac {3 \, a^{2} c d^{3}}{x} - \frac {a^{2} d^{3}}{2 \, x^{2}} + \frac {{\left (2 \, b^{2} c^{3} d^{3} x^{3} - 6 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2}}{8 \, x^{2}} - \int -\frac {{\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 12 \, {\left (a b c^{3} d^{3} x^{3} - a b c^{2} d^{3} x^{2}\right )} \log \left (c x + 1\right ) - {\left (2 \, b^{2} c^{4} d^{3} x^{4} + 12 \, a b c^{3} d^{3} x^{3} - b^{2} c d^{3} x - 6 \, {\left (2 \, a b c^{2} d^{3} + b^{2} c^{2} d^{3}\right )} x^{2} + 2 \, {\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \, {\left (c x^{4} - x^{3}\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3}{x^3} \,d x \]
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 \[ d^{3} \left (\int a^{2} c^{3}\, dx + \int \frac {a^{2}}{x^{3}}\, dx + \int \frac {3 a^{2} c}{x^{2}}\, dx + \int \frac {3 a^{2} c^{2}}{x}\, dx + \int b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int 2 a b c^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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