Optimal. Leaf size=38 \[ -\frac{6 a x+i}{210 a^3 c^{19} (1-i a x)^{21} (1+i a x)^{15}} \]
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Rubi [A] time = 0.0785909, 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} \[ -\frac{6 a x+i}{210 a^3 c^{19} (1-i a x)^{21} (1+i a x)^{15}} \]
Antiderivative was successfully verified.
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Rule 5082
Rule 81
Rubi steps
\begin{align*} \int \frac{e^{6 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^{19}} \, dx &=\frac{\int \frac{x^2}{(1-i a x)^{22} (1+i a x)^{16}} \, dx}{c^{19}}\\ &=-\frac{i+6 a x}{210 a^3 c^{19} (1-i a x)^{21} (1+i a x)^{15}}\\ \end{align*}
Mathematica [A] time = 1.05516, size = 36, normalized size = 0.95 \[ \frac{6 a x+i}{210 a^3 c^{19} (a x-i)^{15} (a x+i)^{21}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.342, size = 34, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{19} \left ( ax+i \right ) ^{21} \left ( ax-i \right ) ^{15}} \left ({\frac{x}{35\,{a}^{2}}}+{\frac{{\frac{i}{210}}}{{a}^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.0134, size = 394, normalized size = 10.37 \begin{align*} \frac{20775821962641408 \, a^{7} x^{7} - 121192294782074880 i \, a^{6} x^{6} - 290861507476979712 \, a^{5} x^{5} + 363576884346224640 i \, a^{4} x^{4} + 242384589564149760 \, a^{3} x^{3} - 72715376869244928 i \, a^{2} x^{2} - 3462636993773568 i}{727153768692449280 \,{\left (a^{45} c^{19} x^{42} + 21 \, a^{43} c^{19} x^{40} + 210 \, a^{41} c^{19} x^{38} + 1330 \, a^{39} c^{19} x^{36} + 5985 \, a^{37} c^{19} x^{34} + 20349 \, a^{35} c^{19} x^{32} + 54264 \, a^{33} c^{19} x^{30} + 116280 \, a^{31} c^{19} x^{28} + 203490 \, a^{29} c^{19} x^{26} + 293930 \, a^{27} c^{19} x^{24} + 352716 \, a^{25} c^{19} x^{22} + 352716 \, a^{23} c^{19} x^{20} + 293930 \, a^{21} c^{19} x^{18} + 203490 \, a^{19} c^{19} x^{16} + 116280 \, a^{17} c^{19} x^{14} + 54264 \, a^{15} c^{19} x^{12} + 20349 \, a^{13} c^{19} x^{10} + 5985 \, a^{11} c^{19} x^{8} + 1330 \, a^{9} c^{19} x^{6} + 210 \, a^{7} c^{19} x^{4} + 21 \, a^{5} c^{19} x^{2} + a^{3} c^{19}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 10.8487, size = 1139, normalized size = 29.97 \begin{align*} \frac{6 \, a x + i}{210 \, a^{39} c^{19} x^{36} + 1260 i \, a^{38} c^{19} x^{35} + 14700 i \, a^{36} c^{19} x^{33} - 22050 \, a^{35} c^{19} x^{32} + 70560 i \, a^{34} c^{19} x^{31} - 188160 \, a^{33} c^{19} x^{30} + 151200 i \, a^{32} c^{19} x^{29} - 819000 \, a^{31} c^{19} x^{28} - 58800 i \, a^{30} c^{19} x^{27} - 2257920 \, a^{29} c^{19} x^{26} - 1375920 i \, a^{28} c^{19} x^{25} - 4204200 \, a^{27} c^{19} x^{24} - 4586400 i \, a^{26} c^{19} x^{23} - 5241600 \, a^{25} c^{19} x^{22} - 9129120 i \, a^{24} c^{19} x^{21} - 3783780 \, a^{23} c^{19} x^{20} - 12612600 i \, a^{22} c^{19} x^{19} - 12612600 i \, a^{20} c^{19} x^{17} + 3783780 \, a^{19} c^{19} x^{16} - 9129120 i \, a^{18} c^{19} x^{15} + 5241600 \, a^{17} c^{19} x^{14} - 4586400 i \, a^{16} c^{19} x^{13} + 4204200 \, a^{15} c^{19} x^{12} - 1375920 i \, a^{14} c^{19} x^{11} + 2257920 \, a^{13} c^{19} x^{10} - 58800 i \, a^{12} c^{19} x^{9} + 819000 \, a^{11} c^{19} x^{8} + 151200 i \, a^{10} c^{19} x^{7} + 188160 \, a^{9} c^{19} x^{6} + 70560 i \, a^{8} c^{19} x^{5} + 22050 \, a^{7} c^{19} x^{4} + 14700 i \, a^{6} c^{19} x^{3} + 1260 i \, a^{4} c^{19} x - 210 \, a^{3} c^{19}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13747, size = 429, normalized size = 11.29 \begin{align*} -\frac{358229025 \, a^{14} x^{14} - 5340869100 \, a^{13} i x^{13} - 37114698075 \, a^{12} x^{12} + 159416118225 \, a^{11} i x^{11} + 473088806190 \, a^{10} x^{10} - 1026819468675 \, a^{9} i x^{9} - 1682288472150 \, a^{8} x^{8} + 2115551402250 \, a^{7} i x^{7} + 2054435046125 \, a^{6} x^{6} - 1535397250002 \, a^{5} i x^{5} - 870854759775 \, a^{4} x^{4} + 364307533205 \, a^{3} i x^{3} + 106553746740 \, a^{2} x^{2} - 19571887695 \, a i x - 1710785408}{901943132160 \,{\left (a x - i\right )}^{15} a^{3} c^{19}} + \frac{358229025 \, a^{20} x^{20} + 7555375800 \, a^{19} i x^{19} - 75901131600 \, a^{18} x^{18} - 483051354975 \, a^{17} i x^{17} + 2184946607340 \, a^{16} x^{16} + 7469205450840 \, a^{15} i x^{15} - 20031221295000 \, a^{14} x^{14} - 43177004037300 \, a^{13} i x^{13} + 76013078916950 \, a^{12} x^{12} + 110448380006328 \, a^{11} i x^{11} - 133277726128008 \, a^{10} x^{10} - 133908931763530 \, a^{9} i x^{9} + 111933156213900 \, a^{8} x^{8} + 77492989590120 \, a^{7} i x^{7} - 44041557267624 \, a^{6} x^{6} - 20244576347604 \, a^{5} i x^{5} + 7349182966545 \, a^{4} x^{4} + 2026362494800 \, a^{3} i x^{3} - 396520754280 \, a^{2} x^{2} - 48177926223 \, a i x + 2584181888}{901943132160 \,{\left (a x + i\right )}^{21} a^{3} c^{19}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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