Integrand size = 28, antiderivative size = 65 \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {(i-5 a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (1-i a x)^{10} (1+i a x)^{15} \sqrt {c+a^2 c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {5193, 5190, 82} \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {(-5 a x+i) \sqrt {a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{10} (1+i a x)^{15} \sqrt {a^2 c x^2+c}} \]
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Rule 82
Rule 5190
Rule 5193
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (1+a^2 x^2\right )^{27/2}} \, dx}{c^{13} \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int \frac {x^2}{(1-i a x)^{11} (1+i a x)^{16}} \, dx}{c^{13} \sqrt {c+a^2 c x^2}} \\ & = \frac {(i-5 a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (1-i a x)^{10} (1+i a x)^{15} \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {(1+5 i a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (-i+a x)^{15} (i+a x)^{10} \sqrt {c+a^2 c x^2}} \]
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Time = 0.37 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {\frac {i x}{24 a^{2}}+\frac {1}{120 a^{3}}}{c^{13} \left (a^{2} x^{2}+1\right )^{\frac {19}{2}} \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x -i\right )^{5}}\) | \(50\) |
default | \(-\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (5 i a x +1\right )}{120 \sqrt {a^{2} x^{2}+1}\, c^{14} a^{3} \left (-a x +i\right )^{15} \left (a x +i\right )^{10}}\) | \(57\) |
gosper | \(-\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (-5 a x +i\right ) \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{120 a^{3} \left (i a x +1\right )^{5} \left (a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}}\) | \(58\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (53) = 106\).
Time = 0.36 (sec) , antiderivative size = 496, normalized size of antiderivative = 7.63 \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {{\left (-i \, a^{22} x^{25} - 5 \, a^{21} x^{24} - 40 \, a^{19} x^{22} + 50 i \, a^{18} x^{21} - 126 \, a^{17} x^{20} + 280 i \, a^{16} x^{19} - 160 \, a^{15} x^{18} + 765 i \, a^{14} x^{17} + 105 \, a^{13} x^{16} + 1248 i \, a^{12} x^{15} + 720 \, a^{11} x^{14} + 1260 i \, a^{10} x^{13} + 1260 \, a^{9} x^{12} + 720 i \, a^{8} x^{11} + 1248 \, a^{7} x^{10} + 105 i \, a^{6} x^{9} + 765 \, a^{5} x^{8} - 160 i \, a^{4} x^{7} + 280 \, a^{3} x^{6} - 126 i \, a^{2} x^{5} + 50 \, a x^{4} - 40 i \, x^{3}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}{120 \, {\left (a^{27} c^{14} x^{27} - 5 i \, a^{26} c^{14} x^{26} + a^{25} c^{14} x^{25} - 45 i \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} - 166 i \, a^{22} c^{14} x^{22} - 330 \, a^{21} c^{14} x^{21} - 286 i \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} - 55 i \, a^{18} c^{14} x^{18} - 2013 \, a^{17} c^{14} x^{17} + 825 i \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} + 1980 i \, a^{14} c^{14} x^{14} - 1980 \, a^{13} c^{14} x^{13} + 2508 i \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} + 2013 i \, a^{10} c^{14} x^{10} + 55 \, a^{9} c^{14} x^{9} + 1045 i \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} + 330 i \, a^{6} c^{14} x^{6} + 166 \, a^{5} c^{14} x^{5} + 50 i \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} - i \, a^{2} c^{14} x^{2} + 5 \, a c^{14} x - i \, c^{14}\right )}} \]
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Timed out. \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (53) = 106\).
Time = 0.27 (sec) , antiderivative size = 274, normalized size of antiderivative = 4.22 \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {5 i \, a \sqrt {c} x + \sqrt {c}}{120 \, {\left (a^{28} c^{14} x^{25} - 5 i \, a^{27} c^{14} x^{24} - 40 i \, a^{25} c^{14} x^{22} - 50 \, a^{24} c^{14} x^{21} - 126 i \, a^{23} c^{14} x^{20} - 280 \, a^{22} c^{14} x^{19} - 160 i \, a^{21} c^{14} x^{18} - 765 \, a^{20} c^{14} x^{17} + 105 i \, a^{19} c^{14} x^{16} - 1248 \, a^{18} c^{14} x^{15} + 720 i \, a^{17} c^{14} x^{14} - 1260 \, a^{16} c^{14} x^{13} + 1260 i \, a^{15} c^{14} x^{12} - 720 \, a^{14} c^{14} x^{11} + 1248 i \, a^{13} c^{14} x^{10} - 105 \, a^{12} c^{14} x^{9} + 765 i \, a^{11} c^{14} x^{8} + 160 \, a^{10} c^{14} x^{7} + 280 i \, a^{9} c^{14} x^{6} + 126 \, a^{8} c^{14} x^{5} + 50 i \, a^{7} c^{14} x^{4} + 40 \, a^{6} c^{14} x^{3} + 5 \, a^{4} c^{14} x - i \, a^{3} c^{14}\right )}} \]
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Exception generated. \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 3.43 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int \frac {e^{-5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {c^2\,\sqrt {a^2\,x^2+1}\,{\left (a\,x+1{}\mathrm {i}\right )}^5\,\left (1+a\,x\,5{}\mathrm {i}\right )}{120\,a^3\,{\left (c\,\left (a^2\,x^2+1\right )\right )}^{31/2}} \]
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