Optimal. Leaf size=476 \[ -\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+3 d^2 e \log (x) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b e^2 x \left (1-c^2 x^2\right ) \left (8 c^2 d+e\right )}{32 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b e^2 \sqrt{c^2 x^2-1} \left (8 c^2 d+e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{32 c^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{c x-1} \sqrt{c x+1}} \]
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Rubi [A] time = 1.76026, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 19, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.905, Rules used = {266, 43, 5790, 12, 6742, 1610, 1807, 1584, 459, 321, 217, 206, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}+3 d^2 e \log (x) \left (a+b \cosh ^{-1}(c x)\right )-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b e^2 x \left (1-c^2 x^2\right ) \left (8 c^2 d+e\right )}{32 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b e^2 \sqrt{c^2 x^2-1} \left (8 c^2 d+e\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{32 c^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 5790
Rule 12
Rule 6742
Rule 1610
Rule 1807
Rule 1584
Rule 459
Rule 321
Rule 217
Rule 206
Rule 2328
Rule 2326
Rule 4625
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-(b c) \int \frac{-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{4 x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{4} (b c) \int \frac{-2 d^3+6 d e^2 x^4+e^3 x^6+12 d^2 e x^2 \log (x)}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{4} (b c) \int \left (\frac{-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{12 d^2 e \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{1}{4} (b c) \int \frac{-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx-\left (3 b c d^2 e\right ) \int \frac{\log (x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{\left (3 b c d^2 e \sqrt{1-c^2 x^2}\right ) \int \frac{\log (x)}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{-2 d^3+6 d e^2 x^4+e^3 x^6}{x^2 \sqrt{-1+c^2 x^2}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b d^2 e \sqrt{1-c^2 x^2}\right ) \int \frac{\sin ^{-1}(c x)}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{6 d e^2 x^3+e^3 x^5}{x \sqrt{-1+c^2 x^2}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b d^2 e \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x^2 \left (6 d e^2+e^3 x^2\right )}{\sqrt{-1+c^2 x^2}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (6 i b d^2 e \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}--\frac{\left (b \left (-24 c^2 d e^2-3 e^3\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{x^2}{\sqrt{-1+c^2 x^2}} \, dx}{16 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b d^2 e \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}--\frac{\left (b \left (-24 c^2 d e^2-3 e^3\right ) \sqrt{-1+c^2 x^2}\right ) \int \frac{1}{\sqrt{-1+c^2 x^2}} \, dx}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 i b d^2 e \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}--\frac{\left (b \left (-24 c^2 d e^2-3 e^3\right ) \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1+c^2 x^2}}\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^3 \left (1-c^2 x^2\right )}{2 x \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b e^2 \left (8 c^2 d+e\right ) x \left (1-c^2 x^2\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b e^3 x^3 \left (1-c^2 x^2\right )}{16 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac{3}{2} d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} e^3 x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 b e^2 \left (8 c^2 d+e\right ) \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{c x}{\sqrt{-1+c^2 x^2}}\right )}{32 c^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+3 d^2 e \left (a+b \cosh ^{-1}(c x)\right ) \log (x)-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log (x)}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \text{Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.657731, size = 267, normalized size = 0.56 \[ \frac{1}{4} \left (-6 b d^2 e \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+12 a d^2 e \log (x)-\frac{2 a d^3}{x^2}+6 a d e^2 x^2+a e^3 x^4-\frac{3 b d e^2 \left (c x \sqrt{c x-1} \sqrt{c x+1}+2 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{c^2}-\frac{b e^3 \left (c x \sqrt{c x-1} \sqrt{c x+1} \left (2 c^2 x^2+3\right )+6 \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{8 c^4}+6 b d^2 e \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )+\frac{2 b d^3 \left (c x \sqrt{c x-1} \sqrt{c x+1}-\cosh ^{-1}(c x)\right )}{x^2}+6 b d e^2 x^2 \cosh ^{-1}(c x)+b e^3 x^4 \cosh ^{-1}(c x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.165, size = 296, normalized size = 0.6 \begin{align*}{\frac{a{e}^{3}{x}^{4}}{4}}+{\frac{3\,a{x}^{2}d{e}^{2}}{2}}+3\,a{d}^{2}e\ln \left ( cx \right ) -{\frac{{d}^{3}a}{2\,{x}^{2}}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{3}{x}^{4}}{4}}-{\frac{b{d}^{3}{\rm arccosh} \left (cx\right )}{2\,{x}^{2}}}+{\frac{3\,b{\rm arccosh} \left (cx\right ){x}^{2}d{e}^{2}}{2}}+{\frac{3\,b{d}^{2}e}{2}{\it polylog} \left ( 2,- \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2} \right ) }-{\frac{b{e}^{3}{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{e}^{3}x}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{{d}^{3}b{c}^{2}}{2}}+{\frac{c{d}^{3}b}{2\,x}\sqrt{cx-1}\sqrt{cx+1}}+3\,b{d}^{2}e{\rm arccosh} \left (cx\right )\ln \left ( \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) ^{2}+1 \right ) -{\frac{3\,bd{\rm arccosh} \left (cx\right ){e}^{2}}{4\,{c}^{2}}}-{\frac{3\,bxd{e}^{2}}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{d}^{2}e \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{2}}-{\frac{3\,b{\rm arccosh} \left (cx\right ){e}^{3}}{32\,{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a e^{3} x^{4} + \frac{3}{2} \, a d e^{2} x^{2} + \frac{1}{2} \, b d^{3}{\left (\frac{\sqrt{c^{2} x^{2} - 1} c}{x} - \frac{\operatorname{arcosh}\left (c x\right )}{x^{2}}\right )} + 3 \, a d^{2} e \log \left (x\right ) - \frac{a d^{3}}{2 \, x^{2}} + \int b e^{3} x^{3} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + 3 \, b d e^{2} x \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) + \frac{3 \, b d^{2} e \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} +{\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \operatorname{arcosh}\left (c x\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{3}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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