Integrand size = 26, antiderivative size = 38 \[ \int \frac {e^{4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=-\frac {i+4 a x}{60 a^3 c^9 (1-i a x)^{10} (1+i a x)^6} \]
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Time = 0.06 (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} \[ \int \frac {e^{4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=-\frac {4 a x+i}{60 a^3 c^9 (1-i a x)^{10} (1+i a x)^6} \]
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Rule 82
Rule 5190
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {x^2}{(1-i a x)^{11} (1+i a x)^7} \, dx}{c^9} \\ & = -\frac {i+4 a x}{60 a^3 c^9 (1-i a x)^{10} (1+i a x)^6} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {e^{4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=-\frac {i+4 a x}{60 a^3 c^9 (-i+a x)^6 (i+a x)^{10}} \]
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Time = 0.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {-\frac {i}{60 a^{3}}-\frac {x}{15 a^{2}}}{\left (a x +i\right )^{10} c^{9} \left (a x -i\right )^{6}}\) | \(34\) |
default | \(-\frac {\frac {i}{60 a^{3}}+\frac {x}{15 a^{2}}}{c^{9} \left (a x +i\right )^{10} \left (a x -i\right )^{6}}\) | \(35\) |
gosper | \(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (4 a x +i\right ) \left (i a x +1\right )^{4}}{60 a^{3} \left (a^{2} x^{2}+1\right )^{11} c^{9}}\) | \(49\) |
parallelrisch | \(\frac {i x^{20} a^{17}+10 i x^{18} a^{15}+45 i x^{16} a^{13}+120 i x^{14} a^{11}+210 i x^{12} a^{9}+252 i x^{10} a^{7}+210 i x^{8} a^{5}+120 i x^{6} a^{3}-4 a^{2} x^{5}+60 i x^{4} a +20 x^{3}}{60 c^{9} \left (a^{2} x^{2}+1\right )^{10}}\) | \(110\) |
norman | \(\frac {\frac {i a \,x^{4}}{c}+\frac {x^{3}}{3 c}-\frac {a^{2} x^{5}}{15 c}+\frac {2 i a^{3} x^{6}}{c}+\frac {7 i a^{5} x^{8}}{2 c}+\frac {21 i a^{7} x^{10}}{5 c}+\frac {7 i a^{9} x^{12}}{2 c}+\frac {2 i a^{11} x^{14}}{c}+\frac {3 i a^{13} x^{16}}{4 c}+\frac {i a^{15} x^{18}}{6 c}+\frac {i a^{17} x^{20}}{60 c}}{\left (a^{2} x^{2}+1\right )^{10} c^{8}}\) | \(142\) |
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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 = 0.28 (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. 194 vs. \(2 (32) = 64\).
Time = 0.81 (sec) , antiderivative size = 194, normalized size of antiderivative = 5.11 \[ \int \frac {e^{4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=\frac {- 4 a x - i}{60 a^{19} c^{9} x^{16} + 240 i a^{18} c^{9} x^{15} + 1200 i a^{16} c^{9} x^{13} - 1200 a^{15} c^{9} x^{12} + 2160 i a^{14} c^{9} x^{11} - 3840 a^{13} c^{9} x^{10} + 1200 i a^{12} c^{9} x^{9} - 5400 a^{11} c^{9} x^{8} - 1200 i a^{10} c^{9} x^{7} - 3840 a^{9} c^{9} x^{6} - 2160 i a^{8} c^{9} x^{5} - 1200 a^{7} c^{9} x^{4} - 1200 i a^{6} c^{9} x^{3} - 240 i a^{4} c^{9} x + 60 a^{3} c^{9}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (30) = 60\).
Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 4.08 \[ \int \frac {e^{4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=-\frac {4 \, a^{5} x^{5} - 15 i \, a^{4} x^{4} - 20 \, a^{3} x^{3} + 10 i \, a^{2} x^{2} + i}{60 \, {\left (a^{23} c^{9} x^{20} + 10 \, a^{21} c^{9} x^{18} + 45 \, a^{19} c^{9} x^{16} + 120 \, a^{17} c^{9} x^{14} + 210 \, a^{15} c^{9} x^{12} + 252 \, a^{13} c^{9} x^{10} + 210 \, a^{11} c^{9} x^{8} + 120 \, a^{9} c^{9} x^{6} + 45 \, a^{7} c^{9} x^{4} + 10 \, a^{5} c^{9} x^{2} + a^{3} c^{9}\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (30) = 60\).
Time = 0.27 (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 = 3.92 (sec) , antiderivative size = 160, normalized size of antiderivative = 4.21 \[ \int \frac {e^{4 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx=-\frac {4\,a^5\,x^5-a^4\,x^4\,15{}\mathrm {i}-20\,a^3\,x^3+a^2\,x^2\,10{}\mathrm {i}+1{}\mathrm {i}}{60\,a^{23}\,c^9\,x^{20}+600\,a^{21}\,c^9\,x^{18}+2700\,a^{19}\,c^9\,x^{16}+7200\,a^{17}\,c^9\,x^{14}+12600\,a^{15}\,c^9\,x^{12}+15120\,a^{13}\,c^9\,x^{10}+12600\,a^{11}\,c^9\,x^8+7200\,a^9\,c^9\,x^6+2700\,a^7\,c^9\,x^4+600\,a^5\,c^9\,x^2+60\,a^3\,c^9} \]
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