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 = 0.0775632, antiderivative size = 38, normalized size of antiderivative = 1., 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 5082
Rule 81
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*}
Mathematica [A] time = 0.194454, 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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Maple [B] time = 0.086, size = 218, normalized size = 5.7 \begin{align*}{\frac{1}{{c}^{9}} \left ({\frac{{\frac{21\,i}{8192}}}{{a}^{3} \left ( -ax+i \right ) ^{4}}}+{\frac{{\frac{i}{1280}}}{{a}^{3} \left ( -ax+i \right ) ^{10}}}-{\frac{{\frac{i}{1024}}}{{a}^{3} \left ( -ax+i \right ) ^{8}}}-{\frac{{\frac{7\,i}{6144}}}{{a}^{3} \left ( -ax+i \right ) ^{6}}}-{\frac{{\frac{165\,i}{65536}}}{{a}^{3} \left ( -ax+i \right ) ^{2}}}+{\frac{1}{768\,{a}^{3} \left ( -ax+i \right ) ^{9}}}-{\frac{21}{10240\,{a}^{3} \left ( -ax+i \right ) ^{5}}}+{\frac{11}{4096\,{a}^{3} \left ( -ax+i \right ) ^{3}}}-{\frac{143}{65536\,{a}^{3} \left ( -ax+i \right ) }}+{\frac{{\frac{13\,i}{16384}}}{{a}^{3} \left ( ax+i \right ) ^{4}}}-{\frac{{\frac{i}{12288}}}{{a}^{3} \left ( ax+i \right ) ^{6}}}-{\frac{{\frac{121\,i}{65536}}}{{a}^{3} \left ( ax+i \right ) ^{2}}}-{\frac{7}{20480\,{a}^{3} \left ( ax+i \right ) ^{5}}}+{\frac{11}{8192\,{a}^{3} \left ( ax+i \right ) ^{3}}}-{\frac{143}{65536\,{a}^{3} \left ( ax+i \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.56486, size = 429, normalized size = 11.29 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 110.205, size = 199, normalized size = 5.24 \begin{align*} - \frac{4 a^{121} x - i a^{120}}{60 a^{139} c^{9} x^{16} - 240 i a^{138} c^{9} x^{15} - 1200 i a^{136} c^{9} x^{13} - 1200 a^{135} c^{9} x^{12} - 2160 i a^{134} c^{9} x^{11} - 3840 a^{133} c^{9} x^{10} - 1200 i a^{132} c^{9} x^{9} - 5400 a^{131} c^{9} x^{8} + 1200 i a^{130} c^{9} x^{7} - 3840 a^{129} c^{9} x^{6} + 2160 i a^{128} c^{9} x^{5} - 1200 a^{127} c^{9} x^{4} + 1200 i a^{126} c^{9} x^{3} + 240 i a^{124} c^{9} x + 60 a^{123} c^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11882, size = 204, normalized size = 5.37 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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