Optimal. Leaf size=207 \[ \frac {a^2 (3 c-2 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c-d) (c+d)^3 \sqrt {c^2-d^2} f}+\frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.23, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2841, 2833, 12,
2739, 632, 210} \begin {gather*} \frac {a^2 (3 c-2 d) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f (c-d) (c+d)^3 \sqrt {c^2-d^2}}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 d f (c-d) (c+d)^3 (c+d \sin (e+f x))}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d f (c+d)^2 (c+d \sin (e+f x))^2}+\frac {a^2 (c-d) \cos (e+f x)}{3 d f (c+d) (c+d \sin (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2833
Rule 2841
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx &=\frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a \int \frac {-6 a d-a (c+5 d) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 d (c+d)}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}+\frac {a \int \frac {10 a (c-d) d+a (c-d) (c+6 d) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 (c-d) d (c+d)^2}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))}-\frac {a \int -\frac {3 a (3 c-2 d) (c-d) d}{c+d \sin (e+f x)} \, dx}{6 (c-d)^2 d (c+d)^3}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))}+\frac {\left (a^2 (3 c-2 d)\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 (c-d) (c+d)^3}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))}+\frac {\left (a^2 (3 c-2 d)\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{(c-d) (c+d)^3 f}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))}-\frac {\left (2 a^2 (3 c-2 d)\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{(c-d) (c+d)^3 f}\\ &=\frac {a^2 (3 c-2 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c-d) (c+d)^3 \sqrt {c^2-d^2} f}+\frac {a^2 (c-d) \cos (e+f x)}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a^2 (c+6 d) \cos (e+f x)}{6 d (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^2 \left (c^2+6 c d-10 d^2\right ) \cos (e+f x)}{6 (c-d) d (c+d)^3 f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 1.64, size = 196, normalized size = 0.95 \begin {gather*} \frac {a^2 \cos (e+f x) \left (-\frac {d (1+\sin (e+f x))^2}{(c+d \sin (e+f x))^3}-\frac {(3 c-2 d) \left (\frac {6 \tanh ^{-1}\left (\frac {\sqrt {c-d} \sqrt {1-\sin (e+f x)}}{\sqrt {-c-d} \sqrt {1+\sin (e+f x)}}\right )}{\sqrt {-c-d} \sqrt {c-d}}-\frac {\sqrt {\cos ^2(e+f x)} (4 c+d+(c+4 d) \sin (e+f x))}{(c+d \sin (e+f x))^2}\right )}{2 (c+d)^2 \sqrt {\cos ^2(e+f x)}}\right )}{3 (-c+d) (c+d) f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(522\) vs.
\(2(196)=392\).
time = 0.86, size = 523, normalized size = 2.53 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 653 vs.
\(2 (203) = 406\).
time = 0.40, size = 1395, normalized size = 6.74 \begin {gather*} \left [-\frac {2 \, {\left (a^{2} c^{4} d + 6 \, a^{2} c^{3} d^{2} - 11 \, a^{2} c^{2} d^{3} - 6 \, a^{2} c d^{4} + 10 \, a^{2} d^{5}\right )} \cos \left (f x + e\right )^{3} - 6 \, {\left (a^{2} c^{5} + 6 \, a^{2} c^{4} d - 8 \, a^{2} c^{3} d^{2} - 8 \, a^{2} c^{2} d^{3} + 7 \, a^{2} c d^{4} + 2 \, a^{2} d^{5}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, {\left (3 \, a^{2} c^{4} - 2 \, a^{2} c^{3} d + 9 \, a^{2} c^{2} d^{2} - 6 \, a^{2} c d^{3} - 3 \, {\left (3 \, a^{2} c^{2} d^{2} - 2 \, a^{2} c d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (9 \, a^{2} c^{3} d - 6 \, a^{2} c^{2} d^{2} + 3 \, a^{2} c d^{3} - 2 \, a^{2} d^{4} - {\left (3 \, a^{2} c d^{3} - 2 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) - 12 \, {\left (2 \, a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} - a^{2} c^{2} d^{3} + 2 \, a^{2} d^{5}\right )} \cos \left (f x + e\right )}{12 \, {\left (3 \, {\left (c^{7} d^{2} + 2 \, c^{6} d^{3} - c^{5} d^{4} - 4 \, c^{4} d^{5} - c^{3} d^{6} + 2 \, c^{2} d^{7} + c d^{8}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{9} + 2 \, c^{8} d + 2 \, c^{7} d^{2} + 2 \, c^{6} d^{3} - 4 \, c^{5} d^{4} - 10 \, c^{4} d^{5} - 2 \, c^{3} d^{6} + 6 \, c^{2} d^{7} + 3 \, c d^{8}\right )} f + {\left ({\left (c^{6} d^{3} + 2 \, c^{5} d^{4} - c^{4} d^{5} - 4 \, c^{3} d^{6} - c^{2} d^{7} + 2 \, c d^{8} + d^{9}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{8} d + 6 \, c^{7} d^{2} - 2 \, c^{6} d^{3} - 10 \, c^{5} d^{4} - 4 \, c^{4} d^{5} + 2 \, c^{3} d^{6} + 2 \, c^{2} d^{7} + 2 \, c d^{8} + d^{9}\right )} f\right )} \sin \left (f x + e\right )\right )}}, -\frac {{\left (a^{2} c^{4} d + 6 \, a^{2} c^{3} d^{2} - 11 \, a^{2} c^{2} d^{3} - 6 \, a^{2} c d^{4} + 10 \, a^{2} d^{5}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{2} c^{5} + 6 \, a^{2} c^{4} d - 8 \, a^{2} c^{3} d^{2} - 8 \, a^{2} c^{2} d^{3} + 7 \, a^{2} c d^{4} + 2 \, a^{2} d^{5}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, {\left (3 \, a^{2} c^{4} - 2 \, a^{2} c^{3} d + 9 \, a^{2} c^{2} d^{2} - 6 \, a^{2} c d^{3} - 3 \, {\left (3 \, a^{2} c^{2} d^{2} - 2 \, a^{2} c d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (9 \, a^{2} c^{3} d - 6 \, a^{2} c^{2} d^{2} + 3 \, a^{2} c d^{3} - 2 \, a^{2} d^{4} - {\left (3 \, a^{2} c d^{3} - 2 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) - 6 \, {\left (2 \, a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} - a^{2} c^{2} d^{3} + 2 \, a^{2} d^{5}\right )} \cos \left (f x + e\right )}{6 \, {\left (3 \, {\left (c^{7} d^{2} + 2 \, c^{6} d^{3} - c^{5} d^{4} - 4 \, c^{4} d^{5} - c^{3} d^{6} + 2 \, c^{2} d^{7} + c d^{8}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{9} + 2 \, c^{8} d + 2 \, c^{7} d^{2} + 2 \, c^{6} d^{3} - 4 \, c^{5} d^{4} - 10 \, c^{4} d^{5} - 2 \, c^{3} d^{6} + 6 \, c^{2} d^{7} + 3 \, c d^{8}\right )} f + {\left ({\left (c^{6} d^{3} + 2 \, c^{5} d^{4} - c^{4} d^{5} - 4 \, c^{3} d^{6} - c^{2} d^{7} + 2 \, c d^{8} + d^{9}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{8} d + 6 \, c^{7} d^{2} - 2 \, c^{6} d^{3} - 10 \, c^{5} d^{4} - 4 \, c^{4} d^{5} + 2 \, c^{3} d^{6} + 2 \, c^{2} d^{7} + 2 \, c d^{8} + d^{9}\right )} f\right )} \sin \left (f x + e\right )\right )}}\right ] \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 781 vs.
\(2 (203) = 406\).
time = 0.51, size = 781, normalized size = 3.77 \begin {gather*} \frac {\frac {3 \, {\left (3 \, a^{2} c - 2 \, a^{2} d\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (c^{4} + 2 \, c^{3} d - 2 \, c d^{3} - d^{4}\right )} \sqrt {c^{2} - d^{2}}} + \frac {3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 12 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 9 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 54 \, a^{2} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 24 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 36 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, a^{2} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 72 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 42 \, a^{2} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 24 \, a^{2} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 72 \, a^{2} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 66 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{2} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 54 \, a^{2} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 42 \, a^{2} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, a^{2} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a^{2} c^{6} + 7 \, a^{2} c^{5} d + 6 \, a^{2} c^{4} d^{2} + 2 \, a^{2} c^{3} d^{3}}{{\left (c^{7} + 2 \, c^{6} d - 2 \, c^{4} d^{3} - c^{3} d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{3}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.32, size = 735, normalized size = 3.55 \begin {gather*} -\frac {\frac {-12\,a^2\,c^3+7\,a^2\,c^2\,d+6\,a^2\,c\,d^2+2\,a^2\,d^3}{3\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (c^4-6\,c^3\,d+4\,c\,d^3+2\,d^4\right )}{c\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {2\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (-4\,c^5+2\,c^4\,d-12\,c^3\,d^2+11\,c^2\,d^3+6\,c\,d^4+2\,d^5\right )}{c^2\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (-4\,c^5+3\,c^4\,d-18\,c^3\,d^2+8\,c^2\,d^3+12\,c\,d^4+4\,d^5\right )}{c^2\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (-c^4-18\,c^3\,d+14\,c^2\,d^2+8\,c\,d^3+2\,d^4\right )}{c\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {2\,a^2\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,c^2+2\,d^2\right )\,\left (-12\,c^3+7\,c^2\,d+6\,c\,d^2+2\,d^3\right )}{3\,c^3\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}}{f\,\left (c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (12\,c^2\,d+8\,d^3\right )+c^3+6\,c^2\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+6\,c^2\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )}-\frac {a^2\,\mathrm {atan}\left (\frac {\left (\frac {a^2\,\left (3\,c-2\,d\right )\,\left (-2\,c^4\,d-4\,c^3\,d^2+4\,c\,d^4+2\,d^5\right )}{2\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{3/2}\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}+\frac {a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,c-2\,d\right )}{{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{3/2}}\right )\,\left (-c^4-2\,c^3\,d+2\,c\,d^3+d^4\right )}{3\,a^2\,c-2\,a^2\,d}\right )\,\left (3\,c-2\,d\right )}{f\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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