Integrand size = 14, antiderivative size = 244 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}+\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}+\frac {d^4 x^8}{72 a}+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{630 a^9} \]
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Time = 0.12 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {200, 5033, 1824, 266} \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {d^3 x^6 \left (36 a^2 c-7 d\right )}{378 a^3}+\frac {d^2 x^4 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right )}{1260 a^5}+\frac {d x^2 \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right )}{630 a^7}+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (a^2 x^2+1\right )}{630 a^9}+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)+\frac {d^4 x^8}{72 a} \]
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Rule 200
Rule 266
Rule 1824
Rule 5033
Rubi steps \begin{align*} \text {integral}& = c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)+a \int \frac {c^4 x+\frac {4}{3} c^3 d x^3+\frac {6}{5} c^2 d^2 x^5+\frac {4}{7} c d^3 x^7+\frac {d^4 x^9}{9}}{1+a^2 x^2} \, dx \\ & = c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)+a \int \left (\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x}{315 a^8}+\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^3}{315 a^6}+\frac {\left (36 a^2 c-7 d\right ) d^3 x^5}{63 a^4}+\frac {d^4 x^7}{9 a^2}+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) x}{315 a^8 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = \frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}+\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}+\frac {d^4 x^8}{72 a}+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \int \frac {x}{1+a^2 x^2} \, dx}{315 a^7} \\ & = \frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}+\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}+\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}+\frac {d^4 x^8}{72 a}+c^4 x \cot ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cot ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cot ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cot ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cot ^{-1}(a x)+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{630 a^9} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.87 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {a^2 d x^2 \left (-420 d^3+30 a^2 d^2 \left (72 c+7 d x^2\right )-4 a^4 d \left (1134 c^2+270 c d x^2+35 d^2 x^4\right )+3 a^6 \left (1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right )\right )+24 a^9 x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right ) \cot ^{-1}(a x)+12 \left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{7560 a^9} \]
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Time = 0.52 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00
method | result | size |
parts | \(\frac {d^{4} x^{9} \operatorname {arccot}\left (a x \right )}{9}+\frac {4 c \,d^{3} x^{7} \operatorname {arccot}\left (a x \right )}{7}+\frac {6 c^{2} d^{2} x^{5} \operatorname {arccot}\left (a x \right )}{5}+\frac {4 c^{3} d \,x^{3} \operatorname {arccot}\left (a x \right )}{3}+c^{4} x \,\operatorname {arccot}\left (a x \right )+\frac {a \left (\frac {d \left (\frac {35}{4} a^{6} d^{3} x^{8}+60 a^{6} c \,d^{2} x^{6}+189 a^{6} c^{2} d \,x^{4}+420 a^{6} c^{3} x^{2}-\frac {35}{3} a^{4} d^{3} x^{6}-90 a^{4} c \,d^{2} x^{4}-378 a^{4} c^{2} d \,x^{2}+\frac {35}{2} a^{2} d^{3} x^{4}+180 a^{2} c \,d^{2} x^{2}-35 d^{3} x^{2}\right )}{2 a^{8}}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{10}}\right )}{315}\) | \(245\) |
derivativedivides | \(\frac {\operatorname {arccot}\left (a x \right ) c^{4} a x +\frac {4 a \,\operatorname {arccot}\left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \,\operatorname {arccot}\left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \,\operatorname {arccot}\left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{4} x^{9}}{9}+\frac {210 c^{3} a^{8} d \,x^{2}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-189 c^{2} a^{6} d^{2} x^{2}+30 c \,a^{8} d^{3} x^{6}-45 a^{6} c \,d^{3} x^{4}+\frac {35 d^{4} a^{8} x^{8}}{8}+90 a^{4} c \,d^{3} x^{2}-\frac {35 d^{4} a^{6} x^{6}}{6}+\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{2} x^{2}}{2}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{315 a^{8}}}{a}\) | \(254\) |
default | \(\frac {\operatorname {arccot}\left (a x \right ) c^{4} a x +\frac {4 a \,\operatorname {arccot}\left (a x \right ) c^{3} d \,x^{3}}{3}+\frac {6 a \,\operatorname {arccot}\left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \,\operatorname {arccot}\left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{4} x^{9}}{9}+\frac {210 c^{3} a^{8} d \,x^{2}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-189 c^{2} a^{6} d^{2} x^{2}+30 c \,a^{8} d^{3} x^{6}-45 a^{6} c \,d^{3} x^{4}+\frac {35 d^{4} a^{8} x^{8}}{8}+90 a^{4} c \,d^{3} x^{2}-\frac {35 d^{4} a^{6} x^{6}}{6}+\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{2} x^{2}}{2}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{315 a^{8}}}{a}\) | \(254\) |
parallelrisch | \(\frac {840 x^{9} \operatorname {arccot}\left (a x \right ) a^{9} d^{4}+4320 x^{7} \operatorname {arccot}\left (a x \right ) a^{9} c \,d^{3}+105 d^{4} a^{8} x^{8}+9072 x^{5} \operatorname {arccot}\left (a x \right ) a^{9} c^{2} d^{2}+720 c \,a^{8} d^{3} x^{6}+10080 x^{3} \operatorname {arccot}\left (a x \right ) a^{9} c^{3} d -140 d^{4} a^{6} x^{6}+2268 c^{2} a^{8} d^{2} x^{4}+7560 x \,\operatorname {arccot}\left (a x \right ) a^{9} c^{4}-1080 a^{6} c \,d^{3} x^{4}+5040 c^{3} a^{8} d \,x^{2}+3780 \ln \left (a^{2} x^{2}+1\right ) a^{8} c^{4}+210 d^{4} a^{4} x^{4}-4536 c^{2} a^{6} d^{2} x^{2}-5040 \ln \left (a^{2} x^{2}+1\right ) a^{6} c^{3} d +2160 a^{4} c \,d^{3} x^{2}+4536 \ln \left (a^{2} x^{2}+1\right ) a^{4} c^{2} d^{2}-420 d^{4} a^{2} x^{2}-2160 \ln \left (a^{2} x^{2}+1\right ) a^{2} c \,d^{3}+420 \ln \left (a^{2} x^{2}+1\right ) d^{4}}{7560 a^{9}}\) | \(297\) |
risch | \(-\frac {d^{4} x^{6}}{54 a^{3}}+\frac {d^{4} x^{4}}{36 a^{5}}-\frac {d^{4} x^{2}}{18 a^{7}}+\frac {\ln \left (-a^{2} x^{2}-1\right ) c^{4}}{2 a}+\frac {\ln \left (-a^{2} x^{2}-1\right ) d^{4}}{18 a^{9}}+\frac {\pi \,d^{4} x^{9}}{18}+\frac {\pi \,c^{4} x}{2}+\frac {2 c \,d^{3} x^{6}}{21 a}+\frac {3 c^{2} d^{2} x^{4}}{10 a}+\frac {2 c^{3} d \,x^{2}}{3 a}-\frac {i d^{4} x^{9} \ln \left (-i a x +1\right )}{18}-\frac {i c^{4} x \ln \left (-i a x +1\right )}{2}-\frac {c \,d^{3} x^{4}}{7 a^{3}}-\frac {3 c^{2} d^{2} x^{2}}{5 a^{3}}+\frac {2 c \,d^{3} x^{2}}{7 a^{5}}-\frac {2 \ln \left (-a^{2} x^{2}-1\right ) c^{3} d}{3 a^{3}}+\frac {3 \ln \left (-a^{2} x^{2}-1\right ) c^{2} d^{2}}{5 a^{5}}-\frac {2 \ln \left (-a^{2} x^{2}-1\right ) c \,d^{3}}{7 a^{7}}+\frac {2 \pi c \,d^{3} x^{7}}{7}+\frac {3 \pi \,c^{2} d^{2} x^{5}}{5}+\frac {2 \pi \,c^{3} d \,x^{3}}{3}+\frac {i \left (35 d^{4} x^{9}+180 d^{3} c \,x^{7}+378 c^{2} d^{2} x^{5}+420 c^{3} d \,x^{3}+315 c^{4} x \right ) \ln \left (i a x +1\right )}{630}-\frac {2 i c^{3} d \,x^{3} \ln \left (-i a x +1\right )}{3}-\frac {2 i c \,d^{3} x^{7} \ln \left (-i a x +1\right )}{7}-\frac {3 i c^{2} d^{2} x^{5} \ln \left (-i a x +1\right )}{5}+\frac {d^{4} x^{8}}{72 a}\) | \(413\) |
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Time = 0.27 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.97 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {105 \, a^{8} d^{4} x^{8} + 20 \, {\left (36 \, a^{8} c d^{3} - 7 \, a^{6} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{8} c^{2} d^{2} - 180 \, a^{6} c d^{3} + 35 \, a^{4} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{8} c^{3} d - 378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} - 35 \, a^{2} d^{4}\right )} x^{2} + 24 \, {\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \operatorname {arccot}\left (a x\right ) + 12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{7560 \, a^{9}} \]
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Time = 0.57 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.50 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\begin {cases} c^{4} x \operatorname {acot}{\left (a x \right )} + \frac {4 c^{3} d x^{3} \operatorname {acot}{\left (a x \right )}}{3} + \frac {6 c^{2} d^{2} x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {4 c d^{3} x^{7} \operatorname {acot}{\left (a x \right )}}{7} + \frac {d^{4} x^{9} \operatorname {acot}{\left (a x \right )}}{9} + \frac {c^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} + \frac {2 c^{3} d x^{2}}{3 a} + \frac {3 c^{2} d^{2} x^{4}}{10 a} + \frac {2 c d^{3} x^{6}}{21 a} + \frac {d^{4} x^{8}}{72 a} - \frac {2 c^{3} d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 a^{3}} - \frac {3 c^{2} d^{2} x^{2}}{5 a^{3}} - \frac {c d^{3} x^{4}}{7 a^{3}} - \frac {d^{4} x^{6}}{54 a^{3}} + \frac {3 c^{2} d^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{5 a^{5}} + \frac {2 c d^{3} x^{2}}{7 a^{5}} + \frac {d^{4} x^{4}}{36 a^{5}} - \frac {2 c d^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{7 a^{7}} - \frac {d^{4} x^{2}}{18 a^{7}} + \frac {d^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{18 a^{9}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{4} x + \frac {4 c^{3} d x^{3}}{3} + \frac {6 c^{2} d^{2} x^{5}}{5} + \frac {4 c d^{3} x^{7}}{7} + \frac {d^{4} x^{9}}{9}\right )}{2} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.93 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {1}{7560} \, a {\left (\frac {105 \, a^{6} d^{4} x^{8} + 20 \, {\left (36 \, a^{6} c d^{3} - 7 \, a^{4} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{6} c^{2} d^{2} - 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{6} c^{3} d - 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} - 35 \, d^{4}\right )} x^{2}}{a^{8}} + \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{10}}\right )} + \frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \operatorname {arccot}\left (a x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.42 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\frac {1}{7560} \, {\left (\frac {24 \, {\left (35 \, d^{4} + \frac {180 \, c d^{3}}{x^{2}} + \frac {378 \, c^{2} d^{2}}{x^{4}} + \frac {420 \, c^{3} d}{x^{6}} + \frac {315 \, c^{4}}{x^{8}}\right )} x^{9} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {{\left (105 \, d^{4} + \frac {720 \, c d^{3}}{x^{2}} + \frac {2268 \, c^{2} d^{2}}{x^{4}} - \frac {140 \, d^{4}}{a^{2} x^{2}} + \frac {5040 \, c^{3} d}{x^{6}} - \frac {1080 \, c d^{3}}{a^{2} x^{4}} + \frac {7875 \, c^{4}}{x^{8}} - \frac {4536 \, c^{2} d^{2}}{a^{2} x^{6}} + \frac {210 \, d^{4}}{a^{4} x^{4}} - \frac {10500 \, c^{3} d}{a^{2} x^{8}} + \frac {2160 \, c d^{3}}{a^{4} x^{6}} + \frac {9450 \, c^{2} d^{2}}{a^{4} x^{8}} - \frac {420 \, d^{4}}{a^{6} x^{6}} - \frac {4500 \, c d^{3}}{a^{6} x^{8}} + \frac {875 \, d^{4}}{a^{8} x^{8}}\right )} x^{8}}{a^{2}} + \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{10}} - \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{10}}\right )} a \]
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Time = 0.25 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.96 \[ \int \left (c+d x^2\right )^4 \cot ^{-1}(a x) \, dx=\mathrm {acot}\left (a\,x\right )\,\left (c^4\,x+\frac {4\,c^3\,d\,x^3}{3}+\frac {6\,c^2\,d^2\,x^5}{5}+\frac {4\,c\,d^3\,x^7}{7}+\frac {d^4\,x^9}{9}\right )-x^2\,\left (\frac {\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{a^2}+\frac {6\,c^2\,d^2}{5\,a}}{2\,a^2}-\frac {2\,c^3\,d}{3\,a}\right )-x^6\,\left (\frac {d^4}{54\,a^3}-\frac {2\,c\,d^3}{21\,a}\right )+x^4\,\left (\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{4\,a^2}+\frac {3\,c^2\,d^2}{10\,a}\right )+\frac {\ln \left (a^2\,x^2+1\right )\,\left (315\,a^8\,c^4-420\,a^6\,c^3\,d+378\,a^4\,c^2\,d^2-180\,a^2\,c\,d^3+35\,d^4\right )}{630\,a^9}+\frac {d^4\,x^8}{72\,a} \]
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