Optimal. Leaf size=38 \[ \frac {-4 a x+i}{60 a^3 c^9 (1-i a x)^6 (1+i a x)^{10}} \]
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Rubi [A] time = 0.08, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5082, 81} \[ \frac {-4 a x+i}{60 a^3 c^9 (1-i a x)^6 (1+i a x)^{10}} \]
Antiderivative was successfully verified.
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Rule 81
Rule 5082
Rubi steps
\begin {align*} \int \frac {e^{-4 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^9} \, dx &=\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*}
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Mathematica [A] time = 0.23, size = 36, normalized size = 0.95 \[ \frac {-4 a x+i}{60 a^3 c^9 (a x-i)^{10} (a x+i)^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 171, normalized size = 4.50 \[ -\frac {4 \, a x - i}{60 \, a^{19} c^{9} x^{16} - 240 i \, a^{18} c^{9} x^{15} - 1200 i \, a^{16} c^{9} x^{13} - 1200 \, a^{15} c^{9} x^{12} - 2160 i \, a^{14} c^{9} x^{11} - 3840 \, a^{13} c^{9} x^{10} - 1200 i \, a^{12} c^{9} x^{9} - 5400 \, a^{11} c^{9} x^{8} + 1200 i \, a^{10} c^{9} x^{7} - 3840 \, a^{9} c^{9} x^{6} + 2160 i \, a^{8} c^{9} x^{5} - 1200 \, a^{7} c^{9} x^{4} + 1200 i \, a^{6} c^{9} x^{3} + 240 i \, a^{4} c^{9} x + 60 \, a^{3} c^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 151, normalized size = 3.97 \[ -\frac {2145 \, a^{5} x^{5} + 12540 \, a^{4} i x^{4} - 30030 \, a^{3} x^{3} - 37080 \, a^{2} i x^{2} + 23841 \, a x + 6476 \, i}{983040 \, {\left (a x + i\right )}^{6} a^{3} c^{9}} + \frac {2145 \, a^{9} x^{9} - 21780 \, a^{8} i x^{8} - 99660 \, a^{7} x^{7} + 270480 \, a^{6} i x^{6} + 481446 \, a^{5} x^{5} - 584920 \, a^{4} i x^{4} - 486220 \, a^{3} x^{3} + 265680 \, a^{2} i x^{2} + 84065 \, a x - 9908 \, i}{983040 \, {\left (a x - i\right )}^{10} a^{3} c^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 218, normalized size = 5.74 \[ \frac {\frac {13 i}{16384 a^{3} \left (a x +i\right )^{4}}-\frac {i}{12288 a^{3} \left (a x +i\right )^{6}}-\frac {121 i}{65536 a^{3} \left (a x +i\right )^{2}}-\frac {7}{20480 a^{3} \left (a x +i\right )^{5}}+\frac {11}{8192 a^{3} \left (a x +i\right )^{3}}-\frac {143}{65536 a^{3} \left (a x +i\right )}+\frac {21 i}{8192 a^{3} \left (-a x +i\right )^{4}}+\frac {i}{1280 a^{3} \left (-a x +i\right )^{10}}-\frac {i}{1024 a^{3} \left (-a x +i\right )^{8}}-\frac {7 i}{6144 a^{3} \left (-a x +i\right )^{6}}-\frac {165 i}{65536 a^{3} \left (-a x +i\right )^{2}}+\frac {1}{768 a^{3} \left (-a x +i\right )^{9}}-\frac {21}{10240 a^{3} \left (-a x +i\right )^{5}}+\frac {11}{4096 a^{3} \left (-a x +i\right )^{3}}-\frac {143}{65536 a^{3} \left (-a x +i\right )}}{c^{9}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.40, size = 159, normalized size = 4.18 \[ \frac {-4\,a^5\,x^5-a^4\,x^4\,15{}\mathrm {i}+20\,a^3\,x^3+a^2\,x^2\,10{}\mathrm {i}+1{}\mathrm {i}}{60\,a^{23}\,c^9\,x^{20}+600\,a^{21}\,c^9\,x^{18}+2700\,a^{19}\,c^9\,x^{16}+7200\,a^{17}\,c^9\,x^{14}+12600\,a^{15}\,c^9\,x^{12}+15120\,a^{13}\,c^9\,x^{10}+12600\,a^{11}\,c^9\,x^8+7200\,a^9\,c^9\,x^6+2700\,a^7\,c^9\,x^4+600\,a^5\,c^9\,x^2+60\,a^3\,c^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.49, size = 192, normalized size = 5.05 \[ \frac {4 a x - i}{- 60 a^{19} c^{9} x^{16} + 240 i a^{18} c^{9} x^{15} + 1200 i a^{16} c^{9} x^{13} + 1200 a^{15} c^{9} x^{12} + 2160 i a^{14} c^{9} x^{11} + 3840 a^{13} c^{9} x^{10} + 1200 i a^{12} c^{9} x^{9} + 5400 a^{11} c^{9} x^{8} - 1200 i a^{10} c^{9} x^{7} + 3840 a^{9} c^{9} x^{6} - 2160 i a^{8} c^{9} x^{5} + 1200 a^{7} c^{9} x^{4} - 1200 i a^{6} c^{9} x^{3} - 240 i a^{4} c^{9} x - 60 a^{3} c^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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