Integrand size = 14, antiderivative size = 53 \[ \int e^{4 i \arctan (a x)} x^2 \, dx=-\frac {8 x}{a^2}-\frac {2 i x^2}{a}+\frac {x^3}{3}-\frac {4}{a^3 (i+a x)}+\frac {12 i \log (i+a x)}{a^3} \]
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Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5170, 90} \[ \int e^{4 i \arctan (a x)} x^2 \, dx=-\frac {4}{a^3 (a x+i)}+\frac {12 i \log (a x+i)}{a^3}-\frac {8 x}{a^2}-\frac {2 i x^2}{a}+\frac {x^3}{3} \]
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Rule 90
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 (1+i a x)^2}{(1-i a x)^2} \, dx \\ & = \int \left (-\frac {8}{a^2}-\frac {4 i x}{a}+x^2+\frac {4}{a^2 (i+a x)^2}+\frac {12 i}{a^2 (i+a x)}\right ) \, dx \\ & = -\frac {8 x}{a^2}-\frac {2 i x^2}{a}+\frac {x^3}{3}-\frac {4}{a^3 (i+a x)}+\frac {12 i \log (i+a x)}{a^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int e^{4 i \arctan (a x)} x^2 \, dx=-\frac {8 x}{a^2}-\frac {2 i x^2}{a}+\frac {x^3}{3}-\frac {4}{a^3 (i+a x)}+\frac {12 i \log (i+a x)}{a^3} \]
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09
method | result | size |
default | \(-\frac {8 x -\frac {1}{3} a^{2} x^{3}+2 i a \,x^{2}}{a^{2}}+\frac {-\frac {4}{a \left (a x +i\right )}+\frac {12 i \ln \left (a x +i\right )}{a}}{a^{2}}\) | \(58\) |
risch | \(-\frac {8 x}{a^{2}}+\frac {x^{3}}{3}-\frac {2 i x^{2}}{a}-\frac {4}{a^{3} \left (a x +i\right )}+\frac {6 i \ln \left (a^{2} x^{2}+1\right )}{a^{3}}+\frac {12 \arctan \left (a x \right )}{a^{3}}\) | \(60\) |
parallelrisch | \(\frac {a^{5} x^{5}-6 i a^{4} x^{4}+36 i \ln \left (a x +i\right ) x^{2} a^{2}-23 a^{3} x^{3}-18 i a^{2} x^{2}+36 i \ln \left (a x +i\right )-36 a x}{3 a^{3} \left (a^{2} x^{2}+1\right )}\) | \(81\) |
meijerg | \(\frac {-\frac {x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2} \left (a^{2} x^{2}+1\right )}+\frac {\left (a^{2}\right )^{\frac {3}{2}} \arctan \left (a x \right )}{a^{3}}}{2 a^{2} \sqrt {a^{2}}}+\frac {2 i \left (-\frac {a^{2} x^{2}}{a^{2} x^{2}+1}+\ln \left (a^{2} x^{2}+1\right )\right )}{a^{3}}-\frac {3 \left (\frac {x \left (a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right )}{5 a^{4} \left (a^{2} x^{2}+1\right )}-\frac {3 \left (a^{2}\right )^{\frac {5}{2}} \arctan \left (a x \right )}{a^{5}}\right )}{a^{2} \sqrt {a^{2}}}-\frac {2 i \left (\frac {x^{2} a^{2} \left (3 a^{2} x^{2}+6\right )}{3 a^{2} x^{2}+3}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{a^{3}}+\frac {-\frac {x \left (a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}+70 a^{2} x^{2}+105\right )}{21 a^{6} \left (a^{2} x^{2}+1\right )}+\frac {5 \left (a^{2}\right )^{\frac {7}{2}} \arctan \left (a x \right )}{a^{7}}}{2 a^{2} \sqrt {a^{2}}}\) | \(254\) |
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.17 \[ \int e^{4 i \arctan (a x)} x^2 \, dx=\frac {a^{4} x^{4} - 5 i \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 24 i \, a x - 36 \, {\left (-i \, a x + 1\right )} \log \left (\frac {a x + i}{a}\right ) - 12}{3 \, {\left (a^{4} x + i \, a^{3}\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int e^{4 i \arctan (a x)} x^2 \, dx=\frac {x^{3}}{3} - \frac {4}{a^{4} x + i a^{3}} - \frac {2 i x^{2}}{a} - \frac {8 x}{a^{2}} + \frac {12 i \log {\left (a x + i \right )}}{a^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.26 \[ \int e^{4 i \arctan (a x)} x^2 \, dx=-\frac {4 \, {\left (a x - i\right )}}{a^{5} x^{2} + a^{3}} + \frac {a^{2} x^{3} - 6 i \, a x^{2} - 24 \, x}{3 \, a^{2}} + \frac {12 \, \arctan \left (a x\right )}{a^{3}} + \frac {6 i \, \log \left (a^{2} x^{2} + 1\right )}{a^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int e^{4 i \arctan (a x)} x^2 \, dx=\frac {12 i \, \log \left (a x + i\right )}{a^{3}} - \frac {4}{{\left (a x + i\right )} a^{3}} + \frac {a^{6} x^{3} - 6 i \, a^{5} x^{2} - 24 \, a^{4} x}{3 \, a^{6}} \]
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Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96 \[ \int e^{4 i \arctan (a x)} x^2 \, dx=\frac {x^3}{3}+\frac {\ln \left (x+\frac {1{}\mathrm {i}}{a}\right )\,12{}\mathrm {i}}{a^3}-\frac {8\,x}{a^2}-\frac {4}{a^4\,\left (x+\frac {1{}\mathrm {i}}{a}\right )}-\frac {x^2\,2{}\mathrm {i}}{a} \]
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