Integrand size = 26, antiderivative size = 38 \[ \int \frac {e^{-4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=\frac {i-4 a x}{60 a^3 c^9 (1-i a x)^6 (1+i a x)^{10}} \]
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Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5190, 82} \[ \int \frac {e^{-4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=\frac {-4 a x+i}{60 a^3 c^9 (1-i a x)^6 (1+i a x)^{10}} \]
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Rule 82
Rule 5190
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {x^2}{(1-i a x)^7 (1+i a x)^{11}} \, dx}{c^9} \\ & = \frac {i-4 a x}{60 a^3 c^9 (1-i a x)^6 (1+i a x)^{10}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=\frac {i-4 a x}{60 a^3 c^9 (-i+a x)^{10} (i+a x)^6} \]
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Time = 0.59 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {\frac {i}{60 a^{3}}-\frac {x}{15 a^{2}}}{c^{9} \left (a x -i\right )^{10} \left (a x +i\right )^{6}}\) | \(34\) |
gosper | \(-\frac {\left (-4 a x +i\right ) \left (a x +i\right ) \left (-a x +i\right )}{60 \left (a^{2} x^{2}+1\right )^{7} c^{9} \left (i a x +1\right )^{4} a^{3}}\) | \(49\) |
parallelrisch | \(-\frac {i x^{16} a^{13}+4 x^{15} a^{12}+20 x^{13} a^{10}-20 i x^{12} a^{9}+36 x^{11} a^{8}-64 i x^{10} a^{7}+20 x^{9} a^{6}-90 i x^{8} a^{5}-20 a^{4} x^{7}-64 i x^{6} a^{3}-36 a^{2} x^{5}-20 i x^{4} a -20 x^{3}}{60 c^{9} \left (-a x +i\right )^{4} \left (a^{2} x^{2}+1\right )^{6}}\) | \(132\) |
norman | \(\frac {-\frac {i a \,x^{4}}{c}+\frac {x^{3}}{3 c}-\frac {a^{2} x^{5}}{15 c}-\frac {2 i a^{3} x^{6}}{c}-\frac {7 i a^{5} x^{8}}{2 c}-\frac {21 i a^{7} x^{10}}{5 c}-\frac {7 i a^{9} x^{12}}{2 c}-\frac {2 i a^{11} x^{14}}{c}-\frac {3 i a^{13} x^{16}}{4 c}-\frac {i a^{15} x^{18}}{6 c}-\frac {i a^{17} x^{20}}{60 c}}{\left (a^{2} x^{2}+1\right )^{10} c^{8}}\) | \(142\) |
default | \(\frac {\frac {21 i}{8192 a^{3} \left (-a x +i\right )^{4}}+\frac {i}{1280 a^{3} \left (-a x +i\right )^{10}}-\frac {i}{1024 a^{3} \left (-a x +i\right )^{8}}-\frac {7 i}{6144 a^{3} \left (-a x +i\right )^{6}}-\frac {165 i}{65536 a^{3} \left (-a x +i\right )^{2}}+\frac {1}{768 a^{3} \left (-a x +i\right )^{9}}-\frac {21}{10240 a^{3} \left (-a x +i\right )^{5}}+\frac {11}{4096 a^{3} \left (-a x +i\right )^{3}}-\frac {143}{65536 a^{3} \left (-a x +i\right )}+\frac {13 i}{16384 a^{3} \left (a x +i\right )^{4}}-\frac {i}{12288 a^{3} \left (a x +i\right )^{6}}-\frac {121 i}{65536 a^{3} \left (a x +i\right )^{2}}-\frac {7}{20480 a^{3} \left (a x +i\right )^{5}}+\frac {11}{8192 a^{3} \left (a x +i\right )^{3}}-\frac {143}{65536 a^{3} \left (a x +i\right )}}{c^{9}}\) | \(218\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 4.45 \[ \int \frac {e^{-4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=-\frac {4 \, a x - i}{60 \, {\left (a^{19} c^{9} x^{16} - 4 i \, a^{18} c^{9} x^{15} - 20 i \, a^{16} c^{9} x^{13} - 20 \, a^{15} c^{9} x^{12} - 36 i \, a^{14} c^{9} x^{11} - 64 \, a^{13} c^{9} x^{10} - 20 i \, a^{12} c^{9} x^{9} - 90 \, a^{11} c^{9} x^{8} + 20 i \, a^{10} c^{9} x^{7} - 64 \, a^{9} c^{9} x^{6} + 36 i \, a^{8} c^{9} x^{5} - 20 \, a^{7} c^{9} x^{4} + 20 i \, a^{6} c^{9} x^{3} + 4 i \, a^{4} c^{9} x + a^{3} c^{9}\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (31) = 62\).
Time = 0.71 (sec) , antiderivative size = 192, normalized size of antiderivative = 5.05 \[ \int \frac {e^{-4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=\frac {- 4 a x + i}{60 a^{19} c^{9} x^{16} - 240 i a^{18} c^{9} x^{15} - 1200 i a^{16} c^{9} x^{13} - 1200 a^{15} c^{9} x^{12} - 2160 i a^{14} c^{9} x^{11} - 3840 a^{13} c^{9} x^{10} - 1200 i a^{12} c^{9} x^{9} - 5400 a^{11} c^{9} x^{8} + 1200 i a^{10} c^{9} x^{7} - 3840 a^{9} c^{9} x^{6} + 2160 i a^{8} c^{9} x^{5} - 1200 a^{7} c^{9} x^{4} + 1200 i a^{6} c^{9} x^{3} + 240 i a^{4} c^{9} x + 60 a^{3} c^{9}} \]
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Exception generated. \[ \int \frac {e^{-4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 3.66 \[ \int \frac {e^{-4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=-\frac {2145 \, a^{5} x^{5} + 12540 i \, a^{4} x^{4} - 30030 \, a^{3} x^{3} - 37080 i \, a^{2} x^{2} + 23841 \, a x + 6476 i}{983040 \, {\left (a x + i\right )}^{6} a^{3} c^{9}} + \frac {2145 \, a^{9} x^{9} - 21780 i \, a^{8} x^{8} - 99660 \, a^{7} x^{7} + 270480 i \, a^{6} x^{6} + 481446 \, a^{5} x^{5} - 584920 i \, a^{4} x^{4} - 486220 \, a^{3} x^{3} + 265680 i \, a^{2} x^{2} + 84065 \, a x - 9908 i}{983040 \, {\left (a x - i\right )}^{10} a^{3} c^{9}} \]
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Time = 3.73 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.18 \[ \int \frac {e^{-4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=\frac {-4\,a^5\,x^5-a^4\,x^4\,15{}\mathrm {i}+20\,a^3\,x^3+a^2\,x^2\,10{}\mathrm {i}+1{}\mathrm {i}}{60\,a^{23}\,c^9\,x^{20}+600\,a^{21}\,c^9\,x^{18}+2700\,a^{19}\,c^9\,x^{16}+7200\,a^{17}\,c^9\,x^{14}+12600\,a^{15}\,c^9\,x^{12}+15120\,a^{13}\,c^9\,x^{10}+12600\,a^{11}\,c^9\,x^8+7200\,a^9\,c^9\,x^6+2700\,a^7\,c^9\,x^4+600\,a^5\,c^9\,x^2+60\,a^3\,c^9} \]
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