Optimal. Leaf size=441 \[ -\frac {14 b^2 f n^2}{e \sqrt {x}}+\frac {2 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right )}{e^2}-\frac {2 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}-\frac {4 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b^2 f^2 n^2 \log (x)}{e^2}+\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac {4 b^2 f^2 n^2 \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {8 b^2 f^2 n^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^2} \]
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Rubi [A]
time = 0.41, antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 17, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {2504, 2442,
46, 2424, 2341, 2422, 2375, 2421, 6724, 2423, 2441, 2352, 2338, 2413, 12, 2339, 30}
\begin {gather*} \frac {4 b f^2 n \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {4 b^2 f^2 n^2 \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{e^2}-\frac {8 b^2 f^2 n^2 \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {f^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}-\frac {2 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}+\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}+\frac {2 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right )}{e^2}-\frac {4 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b^2 f^2 n^2 \log (x)}{e^2}-\frac {14 b^2 f n^2}{e \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 46
Rule 2338
Rule 2339
Rule 2341
Rule 2352
Rule 2375
Rule 2413
Rule 2421
Rule 2422
Rule 2423
Rule 2424
Rule 2441
Rule 2442
Rule 2504
Rule 6724
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx &=-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-(2 b n) \int \left (-\frac {f \left (a+b \log \left (c x^n\right )\right )}{e x^{3/2}}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 x}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 x}\right ) \, dx\\ &=-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+(2 b n) \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx+\frac {(2 b f n) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx}{e}+\frac {\left (b f^2 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{e^2}-\frac {\left (2 b f^2 n\right ) \int \frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{e^2}\\ &=-\frac {8 b^2 f n^2}{e \sqrt {x}}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^3 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt {x}\right ) \sqrt {x}} \, dx}{2 e^2}-\frac {\left (b f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx}{e^2}-\left (2 b^2 n^2\right ) \int \left (-\frac {f}{e x^{3/2}}+\frac {f^2 \log \left (e+f \sqrt {x}\right )}{e^2 x}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right )}{x^2}-\frac {f^2 \log (x)}{2 e^2 x}\right ) \, dx\\ &=-\frac {12 b^2 f n^2}{e \sqrt {x}}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^2}-\frac {\left (2 b f^2 n\right ) \int \frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{e^2}+\left (2 b^2 n^2\right ) \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right )}{x^2} \, dx+\frac {\left (b^2 f^2 n^2\right ) \int \frac {\log (x)}{x} \, dx}{e^2}-\frac {\left (2 b^2 f^2 n^2\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{e^2}\\ &=-\frac {12 b^2 f n^2}{e \sqrt {x}}+\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {f^2 \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\left (4 b^2 n^2\right ) \text {Subst}\left (\int \frac {\log (d (e+f x))}{x^3} \, dx,x,\sqrt {x}\right )-\frac {\left (4 b^2 f^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx}{e^2}-\frac {\left (4 b^2 f^2 n^2\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{e^2}\\ &=-\frac {12 b^2 f n^2}{e \sqrt {x}}-\frac {2 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}-\frac {4 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac {4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {8 b^2 f^2 n^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^2}+\left (2 b^2 f n^2\right ) \text {Subst}\left (\int \frac {1}{x^2 (e+f x)} \, dx,x,\sqrt {x}\right )+\frac {\left (4 b^2 f^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{e^2}\\ &=-\frac {12 b^2 f n^2}{e \sqrt {x}}-\frac {2 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}-\frac {4 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac {4 b^2 f^2 n^2 \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {8 b^2 f^2 n^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^2}+\left (2 b^2 f n^2\right ) \text {Subst}\left (\int \left (\frac {1}{e x^2}-\frac {f}{e^2 x}+\frac {f^2}{e^2 (e+f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {14 b^2 f n^2}{e \sqrt {x}}+\frac {2 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right )}{e^2}-\frac {2 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}-\frac {4 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b^2 f^2 n^2 \log (x)}{e^2}+\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac {4 b^2 f^2 n^2 \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {8 b^2 f^2 n^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^2}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 821, normalized size = 1.86 \begin {gather*} -\frac {3 a^2 e f \sqrt {x}+18 a b e f n \sqrt {x}+42 b^2 e f n^2 \sqrt {x}-3 a^2 f^2 x \log \left (e+f \sqrt {x}\right )-6 a b f^2 n x \log \left (e+f \sqrt {x}\right )-6 b^2 f^2 n^2 x \log \left (e+f \sqrt {x}\right )+3 a^2 e^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+6 a b e^2 n \log \left (d \left (e+f \sqrt {x}\right )\right )+6 b^2 e^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {3}{2} a^2 f^2 x \log (x)+3 a b f^2 n x \log (x)+3 b^2 f^2 n^2 x \log (x)+6 a b f^2 n x \log \left (e+f \sqrt {x}\right ) \log (x)+6 b^2 f^2 n^2 x \log \left (e+f \sqrt {x}\right ) \log (x)-6 a b f^2 n x \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-6 b^2 f^2 n^2 x \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-\frac {3}{2} a b f^2 n x \log ^2(x)-\frac {3}{2} b^2 f^2 n^2 x \log ^2(x)-3 b^2 f^2 n^2 x \log \left (e+f \sqrt {x}\right ) \log ^2(x)+3 b^2 f^2 n^2 x \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+\frac {1}{2} b^2 f^2 n^2 x \log ^3(x)+6 a b e f \sqrt {x} \log \left (c x^n\right )+18 b^2 e f n \sqrt {x} \log \left (c x^n\right )-6 a b f^2 x \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )-6 b^2 f^2 n x \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )+6 a b e^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+6 b^2 e^2 n \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+3 a b f^2 x \log (x) \log \left (c x^n\right )+3 b^2 f^2 n x \log (x) \log \left (c x^n\right )+6 b^2 f^2 n x \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )-6 b^2 f^2 n x \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )-\frac {3}{2} b^2 f^2 n x \log ^2(x) \log \left (c x^n\right )+3 b^2 e f \sqrt {x} \log ^2\left (c x^n\right )-3 b^2 f^2 x \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )+3 b^2 e^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right )+\frac {3}{2} b^2 f^2 x \log (x) \log ^2\left (c x^n\right )-12 b f^2 n x \left (a+b n+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+24 b^2 f^2 n^2 x \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{3 e^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (e +f \sqrt {x}\right )\right )}{x^{2}}\, dx\]
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 [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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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