Optimal. Leaf size=503 \[ \frac {304 b^2 d \sqrt {d-c^2 d x^2}}{3675 c^4}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {152 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11025 c^4}+\frac {38 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{6125 c^4}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^4}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{105 c \sqrt {1-c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{175 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{49 \sqrt {1-c^2 x^2}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2 \]
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Rubi [A]
time = 0.56, antiderivative size = 503, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 14, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {4787, 4783,
4795, 4767, 4715, 267, 4723, 272, 45, 14, 4777, 12, 457, 78} \begin {gather*} -\frac {d x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{35 c^2}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{175 \sqrt {1-c^2 x^2}}+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{105 c \sqrt {1-c^2 x^2}}-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{35 c^4}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{49 \sqrt {1-c^2 x^2}}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {4 b^2 d x \text {ArcSin}(c x) \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^4}+\frac {38 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{6125 c^4}+\frac {304 b^2 d \sqrt {d-c^2 d x^2}}{3675 c^4}+\frac {152 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11025 c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 78
Rule 267
Rule 272
Rule 457
Rule 4715
Rule 4723
Rule 4767
Rule 4777
Rule 4783
Rule 4787
Rule 4795
Rubi steps
\begin {align*} \int x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} (3 d) \int x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{7 \sqrt {1-c^2 x^2}}\\ &=-\frac {2 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{35 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{35 \sqrt {1-c^2 x^2}}-\frac {\left (6 b c d \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {1-c^2 x^2}} \, dx}{7 \sqrt {1-c^2 x^2}}\\ &=-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{35 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (2 b d \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 c \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{245 \sqrt {1-c^2 x^2}}+\frac {\left (6 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1-c^2 x^2}} \, dx}{175 \sqrt {1-c^2 x^2}}\\ &=\frac {2 b d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt {1-c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (2 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2}} \, dx}{105 \sqrt {1-c^2 x^2}}+\frac {\left (4 b d \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x^2 \left (7-5 c^2 x\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{175 \sqrt {1-c^2 x^2}}\\ &=\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt {1-c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{105 \sqrt {1-c^2 x^2}}+\frac {\left (4 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {2}{c^4 \sqrt {1-c^2 x}}+\frac {\sqrt {1-c^2 x}}{c^4}-\frac {8 \left (1-c^2 x\right )^{3/2}}{c^4}+\frac {5 \left (1-c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )}{245 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1-c^2 x}}-\frac {2 \sqrt {1-c^2 x}}{c^4}+\frac {\left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{175 \sqrt {1-c^2 x^2}}\\ &=-\frac {62 b^2 d \sqrt {d-c^2 d x^2}}{1225 c^4}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {74 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3675 c^4}+\frac {38 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{6125 c^4}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^4}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt {1-c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{105 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{35 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {304 b^2 d \sqrt {d-c^2 d x^2}}{3675 c^4}+\frac {4 a b d x \sqrt {d-c^2 d x^2}}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {152 b^2 d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{11025 c^4}+\frac {38 b^2 d \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{6125 c^4}-\frac {2 b^2 d \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{343 c^4}+\frac {4 b^2 d x \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt {1-c^2 x^2}}+\frac {2 b d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{105 c \sqrt {1-c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 \sqrt {1-c^2 x^2}}+\frac {2 b c^3 d x^7 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{49 \sqrt {1-c^2 x^2}}-\frac {2 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^4}-\frac {d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 244, normalized size = 0.49 \begin {gather*} \frac {d \sqrt {d-c^2 d x^2} \left (-11025 a^2 \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right )+210 a b c x \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )+2 b^2 \sqrt {1-c^2 x^2} \left (18692-1679 c^2 x^2-2178 c^4 x^4+1125 c^6 x^6\right )+210 b \left (-105 a \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right )+b c x \left (210+35 c^2 x^2-168 c^4 x^4+75 c^6 x^6\right )\right ) \text {ArcSin}(c x)-11025 b^2 \left (1-c^2 x^2\right )^{5/2} \left (2+5 c^2 x^2\right ) \text {ArcSin}(c x)^2\right )}{385875 c^4 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.43, size = 1678, normalized size = 3.34
method | result | size |
default | \(\text {Expression too large to display}\) | \(1678\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 356, normalized size = 0.71 \begin {gather*} -\frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b^{2} \arcsin \left (c x\right )^{2} - \frac {2}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a b \arcsin \left (c x\right ) - \frac {1}{35} \, {\left (\frac {5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a^{2} + \frac {2}{385875} \, b^{2} {\left (\frac {1125 \, \sqrt {-c^{2} x^{2} + 1} c^{4} d^{\frac {3}{2}} x^{6} - 2178 \, \sqrt {-c^{2} x^{2} + 1} c^{2} d^{\frac {3}{2}} x^{4} - 1679 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}} x^{2} + \frac {18692 \, \sqrt {-c^{2} x^{2} + 1} d^{\frac {3}{2}}}{c^{2}}}{c^{2}} + \frac {105 \, {\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} - 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} + 210 \, d^{\frac {3}{2}} x\right )} \arcsin \left (c x\right )}{c^{3}}\right )} + \frac {2 \, {\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} - 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} + 210 \, d^{\frac {3}{2}} x\right )} a b}{3675 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.92, size = 360, normalized size = 0.72 \begin {gather*} -\frac {210 \, {\left (75 \, a b c^{7} d x^{7} - 168 \, a b c^{5} d x^{5} + 35 \, a b c^{3} d x^{3} + 210 \, a b c d x + {\left (75 \, b^{2} c^{7} d x^{7} - 168 \, b^{2} c^{5} d x^{5} + 35 \, b^{2} c^{3} d x^{3} + 210 \, b^{2} c d x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + {\left (1125 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{8} d x^{8} - 9 \, {\left (15925 \, a^{2} - 734 \, b^{2}\right )} c^{6} d x^{6} + {\left (99225 \, a^{2} - 998 \, b^{2}\right )} c^{4} d x^{4} + {\left (11025 \, a^{2} - 40742 \, b^{2}\right )} c^{2} d x^{2} + 11025 \, {\left (5 \, b^{2} c^{8} d x^{8} - 13 \, b^{2} c^{6} d x^{6} + 9 \, b^{2} c^{4} d x^{4} + b^{2} c^{2} d x^{2} - 2 \, b^{2} d\right )} \arcsin \left (c x\right )^{2} - 2 \, {\left (11025 \, a^{2} - 18692 \, b^{2}\right )} d + 22050 \, {\left (5 \, a b c^{8} d x^{8} - 13 \, a b c^{6} d x^{6} + 9 \, a b c^{4} d x^{4} + a b c^{2} d x^{2} - 2 \, a b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{385875 \, {\left (c^{6} x^{2} - c^{4}\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 x^{3} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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