Optimal. Leaf size=766 \[ \frac {\sin (c+d x) \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{32 b^2 d \sqrt {\sec (c+d x)}}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{192 b^3 d}+\frac {\sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 b^3 d \sqrt {\sec (c+d x)}}+\frac {(a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 a b^3 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{64 b^4 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt {\sec (c+d x)}}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A] time = 2.66, antiderivative size = 766, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ -\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{192 b^3 d}+\frac {\sin (c+d x) \left (5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{32 b^2 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (-2 a^2 b (12 B+5 C)+15 a^3 C+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 b^3 d \sqrt {\sec (c+d x)}}+\frac {(a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (24 a^2 b B-15 a^3 C-4 a b^2 (12 A+7 C)-128 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{192 a b^3 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (-8 a^2 b^2 (2 A+C)+8 a^3 b B-5 a^4 C+32 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{64 b^4 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt {\sec (c+d x)}}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2809
Rule 2816
Rule 2994
Rule 2998
Rule 3049
Rule 3053
Rule 3061
Rule 4221
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {3 a C}{2}+b (4 A+3 C) \cos (c+d x)+\frac {1}{2} (8 b B-5 a C) \cos ^2(c+d x)\right ) \, dx}{4 b}\\ &=\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac {3}{2}}(c+d x)}+\frac {(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{4} a (8 b B-5 a C)+\frac {1}{2} b (16 b B-a C) \cos (c+d x)+\frac {3}{4} \left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{12 b^2}\\ &=\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} a \left (48 A b^2+8 a b B-5 a^2 C+36 b^2 C\right )+\frac {1}{4} b \left (48 A b^2+56 a b B+a^2 C+36 b^2 C\right ) \cos (c+d x)-\frac {1}{8} \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{24 b^2}\\ &=\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt {\sec (c+d x)}}-\frac {\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^3 d}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} a \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right )+\frac {1}{4} a b \left (48 A b^2+8 a b B-5 a^2 C+36 b^2 C\right ) \cos (c+d x)+\frac {3}{8} \left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^3}\\ &=\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt {\sec (c+d x)}}-\frac {\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^3 d}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} a \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right )+\frac {1}{4} a b \left (48 A b^2+8 a b B-5 a^2 C+36 b^2 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^3}+\frac {\left (\left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{128 b^3}\\ &=-\frac {\sqrt {a+b} \left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt {\sec (c+d x)}}-\frac {\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^3 d}+\frac {\left (a \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{384 b^3}+\frac {\left (a \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{384 b^3}\\ &=\frac {(a-b) \sqrt {a+b} \left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 a b^3 d \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b} \left (15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 b^3 d \sqrt {\sec (c+d x)}}-\frac {\sqrt {a+b} \left (8 a^3 b B+32 a b^3 B-5 a^4 C-8 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \Pi \left (\frac {a+b}{b};\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^4 d \sqrt {\sec (c+d x)}}+\frac {C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{4 b d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 A b^2-8 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b^2 d \sqrt {\sec (c+d x)}}+\frac {(8 b B-5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b^2 d \sqrt {\sec (c+d x)}}-\frac {\left (24 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (12 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \sin (c+d x)}{192 b^3 d}\\ \end {align*}
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Mathematica [A] time = 15.14, size = 852, normalized size = 1.11 \[ \frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {(8 b B+a C) \sin (c+d x)}{96 b}+\frac {\left (-5 C a^2+8 b B a+48 A b^2+48 b^2 C\right ) \sin (2 (c+d x))}{192 b^2}+\frac {(8 b B+a C) \sin (3 (c+d x))}{96 b}+\frac {1}{32} C \sin (4 (c+d x))\right )}{d}-\frac {-b (b-a) (a+b) \left (15 C a^3-24 b B a^2+4 b^2 (12 A+7 C) a+128 b^3 B\right ) E\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {b-a}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )+a (a-b) (a+b) \left (15 C a^3-6 b (4 B+5 C) a^2+4 b^2 (12 A+12 B+11 C) a-8 b^3 (12 A+16 B+9 C)\right ) F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {b-a}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )-3 (a-b) \left (5 C a^4-8 b B a^3+8 b^2 (2 A+C) a^2-32 b^3 B a-16 b^4 (4 A+3 C)\right ) \left ((a-b) F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {b-a}{a+b}\right )+2 b \Pi \left (-1;\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {b-a}{a+b}\right )\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{a+b}} \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )-(a-b) b \left (15 C a^3-24 b B a^2+4 b^2 (12 A+7 C) a+128 b^3 B\right ) \tan \left (\frac {1}{2} (c+d x)\right ) \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )-1\right ) \left (a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b\right )}{192 (a-b) b^4 d \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )-1\right ) \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right ) \sqrt {\frac {\tan ^2\left (\frac {1}{2} (c+d x)\right )+1}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \sqrt {\frac {a \tan ^2\left (\frac {1}{2} (c+d x)\right )-b \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}{\tan ^2\left (\frac {1}{2} (c+d x)\right )+1}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 87.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.92, size = 5307, normalized size = 6.93 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {a + b \cos {\left (c + d x \right )}} \left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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